Clinical Trial of Safety, Tolerability and Antitumor Activity of Genetically Engineered T Cells in Combination With Anti-Cancer Agents in Relapsed and Refractory Synovial Sarcoma Expressing New York Esophageal Antigen-1 (NY-ESO-1) and/or LAGE-1a

Inclusion Criteria: * Target Expression Screening Phase: Subject must be \>=10 years of age at the time of signing the informed consent. Younger subjects may only be enrolled upon consultation with Sponsor. * Pathologically or histologically confirmed advanced synovial sarcoma that is either metastatic or unresectable and is either currently being treated with or has completed at least one line of standard chemotherapy regimen or subject is intolerant to it. Subjects can also undergo leukapheresis prior to initiating first line or standard therapy. * Radiologically detectable disease. * Life expectancy \>=3 months. * In the investigator's opinion, the subject is suitable for leukapheresis and, subsequently, for GSK3377794 and pembrolizumab infusion. * An archival biopsy or fresh biopsy (if collected as part of standard of care) of tumor tissue is required to perform antigen expression analysis. * Male or female subjects will be included. Contraception requirements will apply at the time of leukapharesis and treatment. * Capable of giving signed informed consent which includes compliance with the requirements and restrictions listed in the informed consent form (ICF) and protocol. For subjects \<18 years of age (or the legal minimum age in the relevant country) their legal guardian must give informed consent. Pediatric subjects will be included in age-appropriate discussion in order to obtain verbal assent. * Leukapheresis Phase: In addition to the criteria mentioned above, the following criteria must also apply. * Subject's tumor has been reviewed by the GlaxoSmithKline (GSK)-designated laboratory and confirmed as meeting the pre-defined threshold for expression of NY-ESO-1 and/or, if tested, LAGE-1a, using archival or fresh (if taken as standard of care) biopsy. * HLA-A\*02:01, HLA-A\*02:05, and/or HLA-A\*02:06 by high resolution testing. * Left ventricular ejection fraction \>=40%, or Fractional Shortening \>=28%. * Performance status: ECOG of 0-1 or for subjects \<=10 years of age, Lansky \>60. * Subject must have adequate organ function 7 days prior to leukapheresis. * Male or female subjects will be included. Contraceptive use by men or women should be consistent with local regulations regarding the methods of contraception for those participating in clinical studies. a) Male subjects are eligible to participate if they agree to the following during the intervention period starting at the first dose of chemotherapy for at least 12 months after receiving the T-cell infusion, or 4 months after there is no evidence of persistence/ gene modified cells in the subject's blood, whichever is longer. Must use effective contraception for at least 4 months after the last dose of pembrolizumab if this time frame is longer than the duration of contraception required in the context of chemotherapy and gene modified cells. * Refrain from donating sperm and either: Be abstinent from heterosexual or homosexual intercourse as their preferred and usual lifestyle (abstinent on a long term and persistent basis) and agree to remain abstinent OR Must agree to use contraception/barrier: a) Agree to use a male condom and should also be advised of the benefit for a female partner to use a highly effective method of contraception when having sexual intercourse with a woman of childbearing potential (WOCBP) who is not currently pregnant (as a condom may break or leak). b) Agree to use male condom when engaging in any activity that allows for passage of ejaculate to another person. * A female subject is eligible to participate if she is not pregnant or breastfeeding, and at least one of the following conditions applies: a) Is not a WOCBP. OR, b) Is a WOCBP who will agree to use a barrier method (male condom) and use a contraceptive method that is highly effective (with a failure rate of \<1% per year), during the intervention period and for at least 12 months after receiving the T-cell infusion, or 4 months after there is no evidence of persistence/ gene modified cells in the subjects blood, whichever is longer. WOCBP must use effective contraception for at least 4 months after the last dose of pembrolizumab if this time frame is longer than the duration of contraception required in the context of chemotherapy and gene modified cells. WOCBP should also agree not to donate eggs (ova, oocytes) for the purpose of reproduction during this period. The investigator should evaluate the effectiveness of the contraceptive method in relationship to the first dose of study intervention. c) A WOCBP must have a negative highly sensitive pregnancy test (urine or serum as required by local regulations) within 24 hours before any dose of study intervention. If a urine test cannot be confirmed as negative (e.g., an ambiguous result), a serum pregnancy test is required. In such cases, the subject must be excluded from participation if the serum pregnancy result is positive. The investigator is responsible for review of medical history, menstrual history, and recent sexual activity to decrease the risk for inclusion of a woman with an early undetected pregnancy. * Lymphodepletion and Treatment Phase: In addition to the criteria mentioned above, the following criteria must also apply: * Subject has been treated with at least one line of standard chemotherapy regimen and is intolerant of or not actively responding to this regimen (i.e. the subject should not be taken prematurely off their primary regimen if they are continuing to respond to it). * Subject must be presenting with at least two lesions: one measurable per RECIST 1.1 and the other required for biopsy. Lesions situated in a previously irradiated area are considered measurable if progression has been demonstrated in such lesions. * Central nervous system (CNS) metastases with low CNS disease burden are allowed on a case by case basis after risk-benefit evaluation in consultation with the Sponsor Medical Monitor (or designee) if: a) Asymptomatic; b) Clinically stable; c) No history of bleeding within CNS metastases; d) No lesions in the brain stem; e) No untreated leptomeningeal metastases; f) No spinal cord compression; g) Not requiring escalating anti-epileptic treatment; h) Not requiring treatment with steroids; i) If treated with focal radiotherapy (for example, gamma knife radiosurgery), there is at least a 2-week wash-out period; j) Not treated with whole brain radiotherapy within the prior 4 weeks. * Lymphodepleting regimen initiation is allowed if: a) There is clinical and/or radiographic evidence of disease progression; b) If supportive therapy (chemotherapy or radiotherapy) was used, the wash-out periods have been fulfilled; c) Supportive radiotherapy has not affected \>25% of bone marrow; d) Following intermediate standard of care line of therapy. * Another line of standard of care therapy may be administered between leukapheresis and treatment if: a) Indicated based on risk/benefit assessment and/or local regulatory requirements and following agreement with Sponsor's Medical Monitor (or designee); b) Treatments, adverse events and other clinical observations are reported in this study database * A biopsy (excisional, incisional, or core) of tumor tissue obtained following cessation of the last line of treatment for synovial sarcoma but prior to initiating lymphodepleting chemotherapy is mandatory (if it is not feasible to obtain a fresh biopsy, archival tumor tissue is required). Exclusion Criteria: * Target Expression Screening Phase: Prior malignancy other than synovial sarcoma: a) Subjects with a history of completely resected non-melanoma skin cancer or successfully treated in situ carcinoma are eligible. b) Subjects with previous malignancies that have been definitively treated, and have been in remission may be enrolled upon consultation with sponsor Medical Monitor or designee. * Previous treatment with genetically engineered NY-ESO-1 specific T cells. * Previous NY-ESO-1 vaccine. * Prior gene therapy using an integrating vector. * Previous allogeneic hematopoietic stem cell transplant. * Clinically significant systemic illness (e.g. serious active infections or significant cardiac, pulmonary, hepatic or other organ dysfunction), that in the judgment of the Investigator would compromise the subject's ability to tolerate protocol therapy or significantly increase the risk of complications. * Leukapheresis Phase: In addition, to the above criteria subjects are not eligible for leukapharesis if any of the following criteria apply: * Has chronic, recurrent, or active autoimmune/immune-mediated disease that has required systemic treatment in the past 2 years (i.e., with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency) is not considered a form of systemic treatment and is allowed. * Uncontrolled intercurrent illness including, but not limited to: a) Ongoing or active infection requiring systemic therapy; b) Clinically significant cardiac disease defined by congestive heart failure New York Heart Association (NYHA) Class \>1; c) Uncontrolled clinically significant arrhythmia in last 6 months; d) Acute coronary syndrome (angina or myocardial infarction) in last 6 months; e) Severe aortic stenosis or symptomatic mitral stenosis; f) Inadequate pulmonary function with mechanical parameters \<40% predicted (Forced expiratory volume in 1 second \[FEV1\], Forced vital capacity \[FVC\], Total lung capacity \[TLC\], Diffusing capacity of the lung for carbon monoxide \[DLCO\]); g) Interstitial lung disease (subjects with existing pneumonitis as a result of radiation are not excluded; however, subjects cannot be oxygen dependent); h) Any other illness that in the judgment of the Investigator would compromise the subjects ability to tolerate protocol therapy or significantly increase the risk of complications * Current active liver or biliary disease (with the exception of Gilbert's syndrome or asymptomatic gallstones, liver metastases or otherwise stable chronic liver disease per investigator assessment). * Has a history of (non-infectious) pneumonitis that required steroids or has current pneumonitis. * Corrected QT interval duration (QTc) \>450 milliseconds (msec) or QTc \>480 msec for subjects with bundle branch block. The QTc is the QT interval corrected for heart rate according to Bazett's formula (QTcB), Fridericia's formula (QTcF), and/or another method, machine-read or manually over-read. * Subject has a history of allergic reactions attributed to compounds of similar chemical or biologic composition to cyclophosphamide, fludarabine, other agents used in the study (Subjects with vitiligo or resolved childhood asthma/atopy would be an exception to this rule). * Has severe hypersensitivity (\>=Grade 3) to pembrolizumab and/or any of its excipients. * Pregnant or breastfeeding females (due to risk to fetus or newborn). * Any prior treatment-related toxicities must be CTCAE (Version 4.03) \<=Grade 1 at the time of initiating study intervention (except for non-clinically significant toxicities e.g., alopecia, vitiligo). Subjects with Grade 2 toxicities that are deemed stable or irreversible (e.g., peripheral neuropathy) can be enrolled on a case-by-case basis in agreement between the Investigator and Sponsor's Medical Monitor (or designee). * Other standard of care lines of therapy are allowed only if guidelines and wash-out periods are followed. * Investigational treatment within 30 days or 5 half-lives (whichever is shorter) prior to leukapheresis. Investigational vaccines (other than NY-ESO-1 vaccines that are not allowed) must follow the wash-out period. Exceptions to this rule must be evaluated by the Investigator in agreement with the Sponsor's Medical Monitor (or designee). * Presence of hepatitis B surface antigen (HBsAg), positive hepatitis C antibody test result at screening or within 3 months prior to first dose of study intervention. Subjects with positive Hepatitis C antibody due to prior resolved disease are can be enrolled, only if a confirmatory negative Hepatitis C ribonucleic acid (RNA) polymerase chain reaction (PCR) test is obtained. * A positive test for HIV antibody. * Has known psychiatric or substance abuse disorders that would interfere with cooperating with the requirements of the study. * Lymphodepletion and Treatment Phase: In addition to the criteria mentioned above, subjects are not eligible for lymphodepletion or treatment if any of the following criteria apply: Subject has received cytotoxic therapy within 2 weeks prior to lymphodepleting chemotherapy: a. Subjects must have recovered from all AEs due to previous therapies to \<=Grade 1 or Baseline. Subjects with \<=Grade 2 neuropathy are eligible. b. Systemic corticosteroids or any other immunosuppressive therapy within 2 weeks prior to lymphodepleting chemotherapy. * Live vaccine within 30 days prior to the first dose of study drug. Examples of live vaccines include, but are not limited to, the following: measles, mumps, rubella, varicella/zoster (chicken pox), yellow fever, rabies, Bacillus Calmette-Guérin (BCG), and typhoid vaccine. Seasonal influenza vaccines for injection are generally killed virus vaccines and are allowed; however, intranasal influenza vaccines (e.g., FluMist) are live attenuated vaccines and are not allowed. * Major surgery \<=28 days of first dose of study intervention.