Pembrolizumab plus N-803, with or without PD-L1 t‑haNK cells, for operable HPV‑negative head and neck cancer

* INCLUSION CRITERIA:
* Histologically or cytologically confirmed previously untreated intermediate/high risk, p16-negative (if oropharyngeal primary tumor), squamous cell carcinoma of the head and neck (T1-T4, N0-N3, M0 stage II, III or IV). Note: p16 status will be determined by a history of p16 immunohistochemistry (IHC) staining conducted per standard of care.
* Age \>= 18 years.
* ECOG performance status \<= 1.
* Planned for cancer removal surgery per standard of care.
* Participants must have adequate organ and marrow function as defined below:

  * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \<= 3.0 x upper limit of normal (ULN)
  * Total bilirubin \<= 1 x ULN. Note: Participants with Gilbert's syndrome can have total bilirubin \< 3.0 mg/dL
  * Absolute neutrophil count (ANC) \>= 1.0 x 10\^9/L
  * Hemoglobin (Hgb) \>= 10.0 g/dL
  * Platelet count \>= 100 x 10\^9/L
  * Prothrombin time (PT) and partial thromboplastin time (PTT) \<= 1 x ULN. Note: Participants with prolonged PTT determined to be due to lupus anticoagulant are eligible
  * Creatinine \<= 1.5 x ULN
* Participants with a history of human immunodeficiency virus (HIV) infection must:

  * be on effective anti-retroviral therapy; and
  * have the viral load \< 400 copies/mL; and
  * have the CD4 count \> 150 cells/microL
* Participants with a history of Hepatitis C virus (HCV) infection must

  * received curative treatment; and
  * have an undetectable viral load.
* Women of child-bearing potential (WOCBP) must agree to use a highly effective method of contraception (hormonal, intrauterine device (IUD), surgical sterilization) for the duration of the study treatment and up to 4 months after the last dose of the study drug(s).

Men must agree to use an effective method of contraception (barrier, surgical sterilization, abstinence) for the duration of the study treatment and up to 4 months after the last dose of the study drug (s). A participant may request a female partner to use an effective form of contraception to fulfill this requirement. Men must not freeze or donate sperm within the same period.

* Nursing participants must be willing to discontinue nursing from study treatment initiation through 4 months after the last dose of the study drug(s).
* Participants must have a primary tumor site that is amenable to biopsy and be willing to undergo pre-and post-treatment biopsies.
* The ability of a participant to understand and the willingness to sign a written informed consent document.

EXCLUSION CRITERIA:

* History of allergic reactions attributed to compounds of similar chemical or biological composition to drugs used in the study.
* Active immunosuppressive treatment equivalent of \> 10mg of prednisone daily. Note: Short-course systemic corticosteroids (e.g., prevention/treatment for transfusion reaction) or use for a non-cancer indication (e.g., adrenal replacement) is acceptable.
* History of autoimmune disease with the exception of controlled thyroid disease, psoriasis not requiring medications, vitiligo, and alopecia.
* Participants with a history of hepatitis B (HBV).
* Prior malignancy active within the previous 2 years except for locally curable cancer that is currently considered cured and does not require an additional standard of care treatment, such as cutaneous basal or squamous cell carcinoma, superficial bladder cancer, or cervical carcinoma in situ, or an incidental histological finding of prostate cancer.
* Prior therapy with the investigational drug within 2 weeks prior to the treatment initiation.
* Pregnancy (confirmed with Beta-human chorionic gonadotropin (Beta-HCG) serum or urine pregnancy test performed in individuals of childbearing potential at screening).
* Uncontrolled intercurrent illness or medical condition(s) evaluated by medical history and physical exam or situations that are not stable (e.g., recent hospitalization, Emergency Room visit or undergoing medication changes) that the investigator assesses would unacceptably increase of participation in the trial for the participant or impair the ability to evaluate the endpoints of the study.