E7 T-cell Receptor (TCR) -T Cell Induction Therapy for Locoregionally Advanced HPV-associated Cancers

Inclusion Criteria: 1. Histologically confirmed carcinoma of a primary tumor site and stage indicated in Table 3 of the protocol. 2. Tumor with HPV16 genotype as determined by testing performed in a Clinical Laboratory Improvement Amendments (CLIA) certified laboratory. 3. HLA-A\*02:01 allele determined by testing performed in a CLIA certified laboratory. Participants may be enrolled based on low resolution typing (i.e., HLA-A\*02) but the HLA-A\*02:01 allele type must be confirmed prior to apheresis. 4. Measurable disease per RECIST Criteria Version 1.1 or PERCIST. 5. Age \> 18 years. 6. Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 at screening. 7. Negative pregnancy test for women under 55 and all women who have had a menstrual period in the last 12 months. A pregnancy tests is not required for women who have had a bilateral oophorectomy or hysterectomy. 8. Men and women of child-bearing potential must agree to use adequate contraception (i.e., intrauterine device, hormonal barrier method of birth control; abstinence; tubal ligation or vasectomy) prior to study entry and for four months after treatment. Should a women become pregnant or suspect she is pregnant while she is participating in this study, she should inform her treating physician immediately. 9. Seronegative for HIV antibody, hepatitis B surface antigen (sAg), and hepatitis C antibody. If a hepatitis C antibody test is positive, then testing for antigen by reverse transcription polymerase chain reaction (RT-PCR) must be negative. 10. Participants must have organ and marrow function as defined below: 1. Leukocytes \> 3,000/Mantle cell lymphoma (mcL) 2. Absolute neutrophil count \> 1,500/mcL 3. Platelets \> 100,000/mcL 4. Hemoglobin \> 9.0 g/dL 5. Total bilirubin within normal institutional limits except in participants with Gilbert's Syndrome who must have a total bilirubin \< 3.0 mg/dL. 6. Serum aspartate aminotransferase (AST) (SGOT)/ alanine transaminase (ALT)(SGPT) \< 2.5 x upper limit of normal (ULN) 7. Calculated creatinine clearance (CrCl) \>50 mL/min/1.73 m2for participants with creatinine levels above institutional normal (by the Chronic Kidney Disease Epidemiology Collaboration (CKD-EPI) equation). 8. international normalized ratio (INR) or activated partial thromboplastin time ( aPTT) ≤1.5 X ULN unless the subject is receiving anticoagulant therapy. Subjects on anticoagulant therapy must have a PT or aPTT within therapeutic range and no history of severe hemorrhage. 11. Participants must be able to understand and be willing to sign the written informed consent document. 12. Participants must agree to participate in protocol CINJ 192103 (Pro2021002307) for gene therapy long term follow up and in protocol Cancer Institute of New Jersey (CINJ) 192002 (Pro2021000281) for biospecimen collection study. Note: Patients may have undergone minor surgical procedures with the past three weeks, as long as all toxicities have recovered to Grade 1 or less. Exclusion Criteria: Subjects who meet any of the following criteria will be excluded from participation in this study: 1. Prior systemic therapy or definitive chemoradiation for the cancer that is being treated on this protocol. 2. Current treatment with another investigational agent. 3. History of severe allergic reactions to compounds of similar chemical or biological composition to agents used in this study. 4. Uncontrolled intercurrent illness such as active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations at the time of treatment that would limit compliance with study requirements. 5. Subjects with HLA-A\*02:01 damaging mutation or allele loss or other molecular resistance detected by research or clinical sequencing will not be eligible. 6. Documented LVEF of less than or equal to 45% tested. The following participants will undergo cardiac evaluations: 1. Clinically significant atrial and/or ventricular arrhythmias including but not limited to: atrial fibrillation, ventricular tachycardia, second or third degree heart block or 2. Age \> 50 years old 7. Participants with baseline screening pulse oxygen level of \<92% on room air will not be eligible. If the underlying cause of hypoxia improves, they may be reevaluated. 8. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with E7 TCR T cells, breastfeeding should be discontinued if the mother is treated with E7 TCR T cells. These potential risks may also apply to other agents used in this study. 9. Participants with a systemic immunodeficiency including acquired deficiency such as HIV or primary immunodeficiency such as Severe Combined Immunodeficiency Disease are ineligible. The experimental treatment being evaluated in this protocol depends on an intact immune system. Participants who have decreased immune competence may be less responsive to the treatment. 10. Participants on immunosuppressive drugs including corticosteroids unless meeting criteria outlined in Section 6.1 (Prohibited Medications). 11. Participants with potentially severe autoimmune diseases such as Crohn's disease, ulcerative colitis, rheumatoid arthritis, autoimmune hepatitis, autoimmune pancreatitis, or systemic lupus erythematosus are not eligible. Patients with less severe autoimmune diseases such as hypothyroidism, vitiligo, and other minor autoimmune disorders are eligible. 12. Participants with prior or concurrent malignancy whose natural history or treatment is unlikely to interfere with the safety or efficacy assessments of the investigational regimen are eligible for this trial. Examples include, but are not limited to: 1. Carcinoma in situ 2. Cutaneous skin cancers requiring only local excision 3. Low grade non-muscle invasive bladder cancer 4. Low grade prostate cancer 13. Subjects who received a live vaccine within 30 days prior to enrollment are not eligible. 14. Determination by the Principal Investigator that participation is not in the best interest of the research subject or may jeopardize the safety of the subject or integrity of the clinical trial data.