Inclusion Criteria:  Able to understand and voluntarily sign an informed consent form (ICF). Age ≥18 years of age at the time of informed consent. HLA-A*0201 positive by high resolution testing.  Any of the following solid tumors with positive NY-ESO-1 expression characterized by IHC:  Head and neck cancer with ≥85% of tumor cells scored ≥1+. Hepatocellular carcinoma with ≥20% of tumor cells scored ≥1+. Lung squamous cell carcinoma with ≥20% of tumor cells scored ≥1+. Synovial sarcoma with ≥65% of cells scored ≥1+. Triple-negative breast cancer with ≥20% of cells scored ≥1+. Received standard curative or palliative therapy including any targeted therapy based on mutation status for their cancer, or advanced solid tumors for which there is no accepted therapy, standard therapies are no longer effective, or the subject refuses additional standard therapy. Measurable disease as defined per RECIST 1.1 (Eisenhauer 2009). Eastern Cooperative Oncology Group (ECOG) Performance Status ≤1. Life expectancy of >4 months.  Adequate hematologic status as demonstrated by not requiring transfusion support or granulocyte-colony stimulating factor (G-CSF) to maintain:  White blood cell count ≥3,000/mcL. Absolute neutrophil count ≥1,000/mcL. Platelet count ≥100,000/mcL. Hemoglobin ≥9.0 g/dL.  Adequate liver function as demonstrated by:  Serum alanine transaminase (ALT) and aspartate transaminase (AST) ≤3 × upper limit of normal, except in subjects with liver metastasis, who must have AST and ALT ≤5 × upper limit of normal. Total bilirubin ≤1.5 × upper limit of normal, except in subjects with Gilbert's Syndrome, who must have total bilirubin ≤3.0 × upper limit of normal. Creatinine clearance ≥50 mL/min. Creatinine clearance of subjects <65 years of age may be estimated by the Cockcroft-Gault formula. Subjects ≥65 years of age must have renal function measured either by 24-hour urine creatinine collection or by nuclear medicine glomerular filtration rate.  Prior therapies:  Targeted or immunotherapy must be completed at least 4 weeks prior to leukapheresis. Chemotherapy must be completed at least 3 weeks prior to leukapheresis. Systemic corticosteroid or other immunosuppressive therapy must be completed at least 2 weeks prior to leukapheresis. Recovered from previous surgery or treatment-related adverse reactions to <grade 2 CTCAE (except for toxicities such as alopecia or vitiligo or toxicities that may be permanent such as neuropathy) prior to lymphodepletion chemotherapy. Men who are sterile (including vasectomy confirmed by post vasectomy semen analysis) or agree to use a condom with spermicide and to not donate sperm during the study and until the persistence of TCRT-ESO-A2 is no longer detected by qPCR or up to 15 years. Women of non-child bearing potential due to surgical sterilization (at least 6 weeks following surgical bilateral oophorectomy with or without hysterectomy or tubal ligation) confirmed by medical history or menopause (i.e., no menstrual bleeding for more than 12 months in a woman aged ≥45 years), OR women of childbearing potential who test negative for pregnancy at Screening based on serum pregnancy test must be using a highly effective method of contraception from the time of Screening until the persistence of TCRT-ESO-A2 is no longer detected by qPCR or up to 15 years. Note: Highly effective methods of contraception that result in a low failure rate (i.e., <1% per year) when used consistently and correctly include combined (estrogen and progestogen containing) hormonal contraception associated with inhibition of ovulation (oral, intravaginal, or transdermal), progestogen-only hormonal contraception associated with inhibition of ovulation (oral, injectable, or implantable), intrauterine device, intrauterine hormone-releasing system, bilateral tubal occlusion, vasectomized partner, or sexual abstinence. True abstinence, when in line with the preferred and usual lifestyle of the subject, is considered a highly effective method only if defined as refraining from heterosexual intercourse during the entire period of study participation and until the persistence of TCRT-ESO-A2 is no longer detected by qPCR. The reliability of sexual abstinence needs to be evaluated in relation to the duration of the clinical study and the preferred and usual lifestyle of the subject. Periodic abstinence (e.g., calendar, ovulation, symptothermal, and post-ovulation method) and withdrawal are not acceptable methods of contraception.  5.3 Exclusion Criteria  Eligible subjects must not have/be:  Any form of primary immunodeficiency (such as Severe Combined Immunodeficiency Disease or AIDS). Known allergic reactions to any component of the treatments in this study. Echocardiography (ECHO) or multigated acquisition scan (MUGA) indicative of left ventricular ejection fraction (LVEF) ≤45%.  Exclusion Criteria:  Eligible subjects must not have/be:  Any form of primary immunodeficiency (such as Severe Combined Immunodeficiency Disease or AIDS). Known allergic reactions to any component of the treatments in this study. Echocardiography (ECHO) or multigated acquisition scan (MUGA) indicative of left ventricular ejection fraction (LVEF) ≤45%. Pulmonary function test of forced expiratory volume in the first second (FEV1) ≤60% in subjects with a prolonged history of cigarette smoking (e.g. ≥20 pack-year smoking history). Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, unstable cardiac arrhythmia, psychiatric illness, or social situations that would limit compliance with study requirements. Subjects with infection requiring treatment, including but not limited to active tuberculosis, COVID-19, known HIV positive subjects or subjects with clinically active hepatitis A, B and C. Planning to take the following medications within 4 weeks of TCRT-ESO-A2 infusion or during the study: long-term systemic use of steroid hormones, hydroxyurea, immunomodulatory drugs (such as interleukin 2, interferon α or γ, GM-CSF, mTOR inhibitor, cystatin, thymosin, etc.). Recent use of inhaled or topical steroids, topical 5-fluorouracil is not exclusionary, however, use of inhaled steroids before leukapheresis and blood draws should be discouraged. Untreated or symptomatic brain metastasis. History of organ transplantation or allogeneic stem cell transplantation. Known uncontrolled diabetes, pulmonary fibrosis, interstitial lung disease, acute lung disease or liver failure. Previous use of any gene therapy product. Not suitable for enrollment at the discretion of the Principal Investigator.