Safety and Efficacy of Allogeneic HPV-specific T Cells in Adults With Recurrent or Metastatic HPV16+ Cancers

Inclusion Criteria: 1. Have pathologically confirmed incurable, locally recurrent or metastatic HPV16+ HNSCC 2. Male or female ≥ 18 years of age 3. Have an human leukocyte antigen (HLA) partially mismatched (haploidentical) related donor. Acceptable donors include first degree relatives (parent, child, or haploidentical sibling), half-siblings, or second degree relatives (aunt, uncle, cousin, niece, nephew). A patient who has inherited a recombinant haplotype from the parents is eligible if the donor shares at least 1 HLA antigen at each of the HLA-A, HLA-B, and HLA DR isotype (HLA-DR) loci. 4. Prior treatment with a platinum-containing regimen 5. Patients with an FDA-approved indication to receive an anti-programmed cell death protein-1 (PD-1) or anti-programmed death-ligand1 (PD-L1) monoclonal antibody must have received at least one cycle of this therapy prior to receiving treatment on this trial 6. Life expectancy ≥ 4 months at time of screening 7. Measurable disease using RECIST 1.1. Tumor lesions situated in a previously irradiated area are considered measurable if progression has been documented in such lesions 8. Eastern Cooperative Oncology Group (ECOG) performance status of \< 2 (see Appendix A). 9. Adequate organ function per the protocol, as defined below: * Left ventricular ejection fraction \> 35% (within 30 days of eligibility screening) * Total bilirubin \< 3.0 mg/dl unless from Gilbert disease * aspartate aminotransferase (AST) and alanine aminotransferase (ALT) \< 4 x institutional upper limit of normal * Serum creatinine \< 3.0 mg/dl 10. Willing and able to provide written informed consent Exclusion Criteria: 1. Disease that is suitable for local therapy administered with curative intent 2. Requires vasopressor or ventilator support 3. Received antithymocyte globulin or similar anti-T-cell antibody therapy ≤ 4 weeks prior to Cycle 1 Day 1 4. Diagnosis of immunodeficiency or is receiving systemic steroid therapy \>10 mg/day of prednisone or equivalent, or any other form of immunosuppressive therapy within 7 days prior to Cycle 1 Day 1 of study treatment. 5. Active autoimmune disease that has required systemic treatment in past 2 years (ie, with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (eg, thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency) is not considered a form of systemic treatment and is allowed. 6. Active infection requiring systemic therapy 7. History of (non-infectious) pneumonitis that required steroids or current pneumonitis 8. Received any live vaccine for up to 30 days prior to enrollment. 9. Known additional malignancy that is progressing or requires active treatment. Exceptions include basal cell carcinoma of the skin or squamous cell carcinoma of the skin that has undergone potentially curative therapy or in situ cervical cancer not associated with HPV16. 10. Pregnancy or breastfeeding: females of childbearing potential must have a negative serum pregnancy test. 11. Female of childbearing potential or male with a female partner of childbearing potential unwilling to use a highly effective method of contraception (abstinence is acceptable) for the course of the study through 120 days after the last study dose since the effects of this therapy on the developing human fetus are unknown. 12. Inability to comply with study procedures 13. Received chemotherapy or targeted small molecule therapy within 2 weeks of the first dose of cyclophosphamide. Subjects must have recovered (ie, grade ≤ 1 or at baseline) from adverse events (AEs) due to a previously administered agent. Subjects with grade ≤ 2 neuropathy or grade ≤ 2 alopecia are an exception to this criterion. 14. Received prior radiotherapy within 2 weeks of the first dose of cyclophosphamide. Subjects must have recovered from all radiation-related toxicities, not require corticosteroids, and not have had radiation pneumonitis. A 1- week washout is permitted for palliative radiation (≤ 2 weeks of radiotherapy) to non-central nervous system (CNS) disease. 15. Carcinomatous meningitis; and/or active CNS metastases, unless metastases are treated and stable and the subject does not require systemic steroids. 16. Known history of human immunodeficiency virus (HIV), known active hepatitis B virus (HBV; e.g., hepatitis B surface antigen \[HBsAg\] reactive), or hepatitis C virus (HCV; e.g., HCV ribonucleic acid \[RNA\] is detected) 17. Prior treatment with HPV T cells