TBio-4101 and Pembro wi/ or w/o Chemo in Recurrent/Metastatic HNSCC

Inclusion Criteria: * Patients must have pathologically confirmed, recurrent, unresectable or metastatic solid tumors of HNSCC, excluding nasopharyngeal and nasal cavity carcinomas, and must have received no prior treatment for metastatic disease. * Patients must have at least 1 cm3 (1.1 g) of viable tumor tissue amenable for resection for TIL generation, and at least one target tumor that can be used for response assessment by RECIST 1.1 and iRECIST criteria and for mandatory translational tumor biopsy (where applicable). * Patients must be willing and able to undergo an apheresis procedure. * Any systemic therapy, including anti-cancer monoclonal antibodies, must have been completed at least 3 weeks prior to TIL harvest or apheresis, whichever is earlier, and any prior therapy-related adverse events (AEs) must have resolved to Grade ≤ 1 except for alopecia and vitiligo. Neuropathy and anemia must have resolved to Grade ≤ 2. * Age 18-75 years at the time of TIL harvest. * Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1. * Participants given pembrolizumab monotherapy must have a composite positive score ≥ 1. * Demonstrate adequate organ and marrow function. * Seronegative for Human immunodeficiency virus (HIV) antibody, hepatitis B surface antigen, and hepatitis C (HCV) antibody (if HCV antibody positive, must be tested for HCV RNA, which must be negative to be eligible). * The participant (or legally acceptable representative if applicable) provides written informed consent for the trial. * The effects of TBio-4101 on the developing human fetus are unknown. For this reason and because other therapeutic agents used in this trial are known to be teratogenic, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of study drug administration. Exclusion Criteria: * Participants with a known additional malignancy that is progressing or has required active treatment within the past 3 years. * Participants who have received prior cell therapy or prior organ transplant. * Participants who have received any previous treatment with a PD-1 or PD-L1 inhibitor, including but not limited to: nivolumab, atezolizumab, pembrolizumab, avelumab, or durvalumab. * Participants who are receiving any other investigational agents. * Participants who have a history of severe immediate hypersensitivity reaction to the study agents or any of their constituents. * Participants who have a severe allergy to gentamicin (used in the manufacturing of TBio-4101). * Participants who have received a live or live-attenuated vaccine within 30 days prior to TIL harvest or apheresis, whichever is earlier. Note: Administration of killed vaccines is allowed. * Participants who require chronic anti-coagulant therapy that cannot either be discontinued or changed to an anti-coagulant with a relatively short half-life, such as a low molecular weight heparin. * Participants with either a primary immunodeficiency disorder (i.e., severe combined immunodeficiency syndrome) or acquired immunodeficiency disorders (such as HIV/AIDS). * Participants with a diagnosis of immunodeficiency or receiving chronic systemic steroid therapy (in dosing exceeding 10 mg daily of prednisone equivalent) or any other form of immunosuppressive therapy within 7 days prior to TIL harvest or apheresis, whichever is earlier. * Participants with a serious cardiac condition, such as uncontrolled hypertension, concurrent congestive heart failure, prior history of Class III/IV cardiac disease (New York Heart Association \[NYHA\]), left ventricular ejection fraction (LVEF) \<45% at baseline, history of cardiac ischemia within the past 6 months, or prior history of cardiac arrhythmia requiring treatment, unless approved by the Sponsor (MCC). Participants who are \> 60 years of age must undergo cardiology clearance exam and cardiac stress test, unless performed within the last 6 months and there has been no clinical change with respect to heart function. * Participants with a forced expiratory volume (FEV1) ≤ 60% of predicted value and diffusing capacity of the lung for carbon monoxide (DLCO) (corrected) \< 60% of predicted value. * Participants with known active central nervous system (CNS) metastases and/or carcinomatous meningitis. Participants with previously treated brain metastases may participate provided they are radiologically stable, i.e., without evidence of progression for at least 4 weeks by repeat imaging (note that the repeat imaging should be performed during study screening), clinically stable and without requirement of steroid treatment for at least 14 days prior to TIL harvest/ apheresis. * Participants with active infections requiring parenteral antibiotics that, in the opinion of the Investigator, would increase the risk of adverse events during TIL harvest or apheresis or start of NMA-LD chemotherapy. * Participants with a history or current evidence of any condition, therapy, or laboratory abnormality, or other circumstance that might confound the results of the study or interfere with the participant's participation for the full duration of the study, such that it is not in the best interest of the patient to participate, in the opinion of the treating Investigator. * Pregnant women are excluded from this study because the effects of TBio-4101 on the developing human fetus are unknown. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with TBio-4101, breastfeeding should be discontinued.