Trial of Cemiplimab, or Cemip-Chemo Followed by Biomarker-guided Treatment for Pts w/HPV H&N Ca

Inclusion Criteria: * Subjects must have pathologically confirmed HPV-positive head and neck squamous cell carcinoma of the oropharynx. Confirmed HPV-positive disease of other subsites are uncommon but also eligible. * HPV testing must be compliant with the following criteria: * p16 IHC positivity is sufficient to enroll and initiate treatment (p16 IHC interpretation to follow guidelines by Jordan and Lingen et al89). * p16 IHC positivity is to be validated using an HPV nucleic acid based secondary assay (HPV ISH, HPV PCR, HPV cfDNA) before or during the neoadjuvant phase.\* * HPV DNA ISH is acceptable if positive, however a negative HPV DNA ISH should be confirmed by HPV RNA ISH or other nucleic acid based method. * HPV16 type (non-HPV16 related cancers are not eligible)\* * \*In the rare event that a subject starts treatment based on p16 IHC alone and HPV type determination is not yet available, subject may commence neoadjuvant treatment based on p16 IHC alone, as along as HPV nucleic acid testing is pending. Patients with non-HPV16 associated tumors will have to leave the study. Given the prevalence of HPV16 (\~90-95%) and usual rapid turnaround of HPV16 RNA-ISH (other assays) this is not expected, but the primary goal is not to have unnecessary treatment delay for subjects. * Availability of ≥8 unstained 5 micron slides. Subjects who cannot fulfill this requirement will need to undergo a new biopsy prior to enrollment on study. In patients where biopsy is not safe, or logistically feasible this requirement can be waived by the PI or a lower number of slides can be accepted. * Subjects must be at least 18 years of age. * AJCC 7th edition: Stage III, IV without bulky N2b/c disease (defined as N3 equivalent volume) and without bulky T4 (≤30cc). (AJCC 8th edition: Stage II or III, or stage I with N1 or N2 nodes (\>3cm or multiple), without bulky nodal disease (defined as N3 equivalent volume) and without bulky T4 disease (defined as 30cc tumor volume)). * Measurable disease (either primary site and/or nodal disease) by RECIST 1.1 criteria. * No previous radiation or chemotherapy for a head and neck cancer. * No complete surgical resection for a head and neck cancer within 8 weeks of enrollment (although lymph node biopsy including excision of an individual node with presence of residual nodal disease, or surgical biopsy/excision of the tumor with residual disease is acceptable). * ECOG performance status 0-1 (Karnofsky ³70%). * Normal Organ Function * Leukocytes ≥2500/mm3, * Platelets ≥75,000/mm3, * Absolute neutrophil count ≥1,500, * Hemoglobin \>9.0 gm/dL, * AST and ALT \<2.5 X ULN * Alkaline phosphatase \<2.5 X ULN * Albumin \>2.9 gm/dL, * Total bilirubin ≤1.5 mg/dl, * Creatinine clearance \>45 mL/min (or SCr \<1.6 mg/dL) within 4 weeks prior to start of treatment. * The standard Cockcroft and Gault formula or the measured glomerular filtration rate must be used to calculate CrCl for enrollment or dosing * Subjects must sign a study-specific informed consent form prior to study entry. Subjects should have the ability to understand and the willingness to sign a written informed consent document. * Sex, and Reproductive Status: * Women of childbearing potential (WOCBP=premenopausal woman capable of becoming pregnant) must have a negative serum or urine pregnancy test (minimum sensitivity 25 IU/L or equivalent units of HCG) within 24 hours prior to the start of study drug. * Women must not be breastfeeding. * WOCBP must agree to follow instructions for method(s) of contraception for the duration of treatment with study drug(s) plus 5 half-lives of study drug(s) plus 30 days (duration of ovulatory cycle) for up to 5 months post-treatment completion. * Men who are sexually active with WOCBP must agree to follow instructions for method(s) of contraception for the duration of treatment with study drug(s) plus 5 half-lives of study drug(s) plus 90 days (duration of sperm turnover) for up to 7 months post treatment completion. * Azoospermic males and WOCBP who are continuously not heterosexually active are exempt from contraceptive requirements. However, they must still undergo pregnancy testing as described in this section. Exclusion Criteria: * Unequivocal demonstration of distant metastases (M1 disease). * Unidentifiable/unknown primary site (neither imaging nor exam nor biopsy can identify the primary). Treating physicians should agree that the primary is sufficiently identified to proceed with clinical care/treatment (e.g. in the case of imaging localization, but absence of biopsy proven pathology) * Intercurrent medical illnesses which would impair subject tolerance to therapy or limit survival. Including but not limited to ongoing or active infection, immunodeficiency, symptomatic congestive heart failure, pulmonary dysfunction, cardiomyopathy, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance. Once clinically stable, as defined by the PI, they are eligible. * Pregnant and nursing women are excluded because of the potential teratogenic effects and potential unknown effects on nursing newborns (please see above paragraph under inclusion criteria regarding WOCBP) * Prior surgical therapy other than incisional/excisional biopsy or organ-sparing procedures such as debulking of airway-compromising tumors. Residual measurable tumor is required for enrollment on study as outlined above * Subjects receiving other investigational agents. * Peripheral neuropathy \>grade 1 * Has a diagnosis of immunodeficiency or is receiving systemic steroid therapy in excess of physiologic dose or any other form of immunosuppressive therapy within 7 days prior to the first dose of trial treatment. * Has a known history of active tuberculosis (Bacillus Tuberculosis infection) * Has hypersensitivity to cemiplimab or any other drug used in this protocol. * Has had a prior systemic anti-cancer treatment within the last 8 weeks * Has a 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 or thyroid cancers, any tumors that are not likely to influence life expectancy in the subsequent 3 years without active treatment other than hormonal therapies (e.g., adjuvant after breast cancer, or low grade prostate cancer). * Has active autoimmune disease that has required systemic treatment in the past year (i.e., with use of steroids or immunosuppressive drugs). Replacement therapy e.g., levothyroxine, insulin, or physiologic corticosteroid doses for adrenal or pituitary insufficiency, etc. are not considered a form of systemic treatment. * Has known history of, or any evidence of active, non-infectious pneumonitis. * Has a history of Human Immunodeficiency Virus (HIV) (HIV ½ antibodies). * Has known active Hepatitis B (e.g., HbsAg reactive) or Hepatitis C (e.g., HCV RNA \[qualitative\] is detected). However, if eradicated subject is eligible. * Has received a live vaccine within 28 days of planned start of study therapy. * Note: Seasonal influenza vaccines for injection are generally inactivated flu vaccines and are allowed; however intranasal influenza vaccines (e.g., Flu-Mist®) are live attenuated vaccines, and are not allowed within 28 days prior to initiation of treatment.