Anti-Mesothelin TNaive/SCM hYP218 (TNhYP218) CAR T Cells in Participants With Mesothelin-Expressing Solid Tumors Including Mesothelioma

* INCLUSION CRITERIA: In order to be eligible to participate in this study, an individual must meet all of the following criteria. For this protocol, treatment initiation is defined as the first day of lymphodepleting chemotherapy. * Participant must have unresectable, locally advanced, or metastatic, or recurrent mesothelioma and other mesothelin expressing solid tumors. For participants with mesothelioma only those with epithelioid or biphasic histology (with \>80% epithelioid component) will be eligible. The diagnosis will be confirmed by the Laboratory of Pathology, CCR, NCI. * Participant must have progressed on at least one FDA-approved systemic therapy considered standard of care for their tumor type. There is no limit on the number of prior treatment regimens. Note: Given the aggressive nature of pancreatic cancer, otherwise eligible individuals with this cancer type can undergo leukapheresis before or while they are getting their frontline treatment as long as they meet all other inclusion criteria. However, TNhYP218 CAR T cells will only be administered after progression on first line standard of care therapy. * Participant must have at least 1 measurable lesion by RECIST version 1.1. * Tumor must have MSLN positivity of 2+ to 3+ in \>= 50% cancer cells by immunohistochemistry on freshly collected biopsy or archival tissue. * Age \>= 18 years. * Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1. * Participants must have adequate organ and marrow function as defined below: System: Laboratory Value Hematological * Hemoglobi: \>=9 g/dL(a) * absolute neutrophil count: \>=1,500/mcL * platelets: \>=100,000/mcL Hepatic * total bilirubin: \<=2.5 X institutional ULN OR direct bilirubin ULN for participants with total bilirubin levels \>1.5 X ULN * AST and ALT \<= 2.5 X institutional ULN (\<= 5 X ULN for participants with liver metastases) Renal * Creatinine OR: \<=1.5 X ULN OR * Calculated(b) creatinine clearance (GFR can also be used in place of creatinine or CrCl) \>= 50 mL/min for participant with creatinine levels \> 1.5 X institutional ULN Coagulation * International normalized ratio (INR) OR prothrombin time (PT): \<=1.5 X ULN unless participant is receiving anticoagulant therapy if PT or aPTT is within therapeutic range of intended use of anticoagulants * Activated partial thromboplastin time (aPTT): \<=1.5 X ULN unless participant is receiving anticoagulant therapy if PT or aPTT is within therapeutic range of intended use of anticoagulants ALT (SGPT)=alanine aminotransferase (serum glutamic pyruvic transaminase); AST (SGOT)=aspartate aminotransferase (serum glutamic oxaloacetic transaminase); GFR=glomerular filtration rate; ULN=upper limit of normal. 1. Criteria must be met without erythropoietin dependency and without packed red blood cell (pRBC) transfusion within last 2 weeks. 2. Creatinine clearance (CrCl) should be calculated per institutional standard. * Normal cardiac ejection fraction (\>= 45% by echocardiogram) and no evidence of hemodynamically significant pericardial effusion as determined by an echocardiogram. * Room air oxygen saturation of 90% or greater. * Treatment-related toxicities from prior treatments must be resolved to \<= grade 2. * Participants with CNS metastases, leptomeningeal disease or carcinomatous meningitis are eligible if they are asymptomatic, have completed their treatment for CNS disease and have recovered from the acute effects of radiation therapy or surgery prior to study entry. Participants must have radiographically stable CNS disease without associated edema at least three months prior to study entry. Additionally, participants have had to have discontinued corticosteroid treatment or non-prophylactic antiseizure medications for these metastases at least four weeks prior to study entry. * Participants of child-bearing potential and participants who can father children must agree to use highly effective contraception or abstinence. * Participants who are nursing or plan to nurse a child must agree to discontinue/postpone nursing for the duration of study therapy and for 12 months after the administration of the cell product or for 4 months from the time no evidence of persistence/gene modified cells is documented in the participant s blood. * Ability of participant to understand and the willingness to sign a written informed consent document. EXCLUSION CRITERIA: An individual who meets any of the following criteria will be excluded from participation in this study: * Prior systemic therapy, an investigational therapy, radiation, and/or surgery within 14 days prior to leukapheresis and 21 days prior to lymphodepleting chemotherapy. * Prior administration of anti-PD-1 or anti-PD-L1 antibodies or other agents that in the opinion of the PI can stimulate immune activity and interfere with an infusion of CAR-T cells within 8 weeks prior to treatment initiation. * Participants with any form of primary immunodeficiency (e.g. severe combined immunodeficiency). * Participants with active or history of autoimmune or immune mediated disease such as multiple sclerosis, lupus, inflammatory bowel disease, rheumatoid arthritis, or small vessel vasculitis. NOTE: Participants with vitiligo, endocrine deficiencies including thyroiditis managed with replacement hormones including physiologic corticosteroids are eligible. * History of severe immediate hypersensitivity reaction to cyclophosphamide or fludarabine. * Therapeutic doses of systemic corticosteroid therapy within 14 days prior to treatment initiation. Physiological doses of steroids (up to 5mg/day of prednisolone or equivalent) are allowed. Corticosteroid creams, ointments, and eye drops are allowed. * Participants with lung fibrosis, inflammatory lung disease or evidence of pneumonitis on baseline imaging studies or medical history of these disorders. * Participant has any other prior or concurrent malignancy with the following exceptions: * Adequately treated basal cell or squamous cell carcinoma * In situ carcinoma of the cervix or breast, treated curatively and without evidence of recurrence for at least 12 months prior to initiation of study therapy. * Treated non-melanoma skin cancer. * Stage 0 or 1 melanoma completely resected at least 12 months prior to initiation of study therapy. * Successfully treated organ-confined prostate cancer with no evidence of progressive disease based on PSA levels and are not on active therapy. * A primary malignancy which has been completely resected and in complete remission for \>= 5 years. * Electrocardiogram showing a QTc interval \> 450 msec in males and \> 470 msec in females (\> 80 msec for participants with bundle branch block). Either Fridericia s or Bazett s formula may be used to correct the QT interval. * Participant has active infection with HIV, hepatitis B virus, HCV, or HTLV as defined below: * Positive serology for HIV, HTLV-1, or HTLV-2. * Active hepatitis B infection as demonstrated by test for hepatitis B surface antigen. Participants who are hepatitis B surface antigen negative but are hepatitis B core antibody positive must have undetectable hepatitis B DNA and receive prophylaxis against viral reactivation. * Active hepatitis C infection as demonstrated by hepatitis C RNA test. Participants who are HCV antibody positive will be screened for HCV RNA by any reverse transcription PCR or branched DNA assay. If HCV antibody is positive, eligibility will be determined based on a negative screening RNA value. * Participant is pregnant or intends to be pregnant during the required period of contraception for participants of childbearing potential. * Participants who received live or attenuated vaccine or virus-based vaccine within 30 days before initiation of treatment initiation * Participants with a history of seizure disorder unless due to now treated metastatic lesions. * Ongoing uncontrolled intercurrent illness, including but not limited to ongoing or active infection, that would impact participant safety or limit compliance with study requirements.