Bempegaldesleukin (NKTR-214) With Radiation and Anti-PD-1 Immunotherapy for Head and Neck Squamous Cell Carcinoma

Inclusion Criteria: * Written informed consent and HIPAA authorization for release of personal health information. NOTE: HIPAA authorization may be included in the informed consent or obtained separately * Qualify for anti-PD-1 therapy based on current guidelines at the time of registration. This includes the standard requirement for the participant's tumor to have been previously determined to express PD-L1 with a combined positive score ≥ 1, as determined by an FDA-approved test. * Eastern Cooperative Oncology Group (ECOG) Performance Status of 0-2 within 30 days prior to enrollment * Histologically proven diagnosis of head and neck squamous cell carcinoma that is metastatic or recurrent disease that is surgically incurable * Prior cancer treatment other than anti-PD-1 therapy must be completed at least 30 days prior to registration and the participant must have recovered from all reversible acute toxic effects of the regimen (other than alopecia) to ≤ Grade 1 or baseline. Participants may not undergo concurrent anti-cancer treatment during treatment with protocol therapies. This includes no treatment with growth factors, tyrosine kinase inhibitors, tumor-specific antibodies, or cytotoxic chemotherapies such as cisplatin. Participants who have previously or are currently taking an anti-PD-1 therapy are eligible for this study if they meet eligibility criteria 2. Participants who have previously taken any other immune checkpoint inhibitor are eligible as long as they have completed that treatment at least 30 days prior to registration and meet all other eligibility criteria. * Participants with central nervous system (CNS) metastases are eligible if the CNS lesions are stable for at least 2 months and if tapered off treatment doses of systemic corticosteroids for at least 2 weeks prior to enrollment on the trial. Management with maintenance physiologic doses of corticosteroids (equivalent doses of prednisone ≤ 10 mg daily) is acceptable. * Participants must have an "index" tumor that: 1) is deemed by the treating radiation oncologist to potentially benefit from palliative radiation 2) is amenable to biopsy, 3) is ≥ 1 cm in longest dimension. * Demonstrate adequate organ function as defined in the table below; all screening labs to be obtained within 30 days prior to enrollment * White blood cell (WBC) ≥ 3,000/mm3 * Absolute Neutrophil Count (ANC) ≥ 1,500/mm3 * Hemoglobin (Hgb) ≥ 9 g/dL * Platelets ≥ 100,000/mm3 * Serum creatinine ≤ 2.0 mg/dL * Total Bilirubin ≤ 2.0 × upper limit of normal (ULN) (\< 3.0 for subjects with Gilbert's Syndrome) * Aspartate aminotransferase (AST) ≤ 3 × ULN * Alanine aminotransferase (ALT) ≤ 3 × ULN * Females of childbearing potential must have a negative serum pregnancy test within 14 days prior to enrollment and must agree to use effective contraception during active treatment and for 5 months after last dose of pembrolizumab and/or NKTR-214. NOTE: Females are considered of childbearing potential unless they are surgically sterile (have undergone a hysterectomy, bilateral tubal ligation, or bilateral oophorectomy) or they are naturally postmenopausal for ≥ 12 consecutive months. * As determined by the enrolling physician or protocol designee, ability of the participant to understand and comply with study procedures for the entire length of the study Exclusion Criteria: * Subjects with significant intercurrent illnesses per physician discretion * Subjects with active or acute infections or active peptic ulcers, unless these conditions are adequately corrected or controlled, in the opinion of the treating physician * Subjects with a diagnosed auto-immune disease (exceptions: subjects with controlled diabetes mellitus type I, thyroid disease, rheumatoid arthritis, vitiligo and alopecia areata not requiring treatment with immunosuppressants are eligible) * Subjects with a history of diabetes mellitus requiring systemic therapy within the past 3 months (i.e. either oral hypoglycemic agents or insulin) must have a documented Hemoglobin A1c \< 8.0 % within 90 days of registration. * Subjects with known genetic conditions causing pre-disposition to radiotherapy (RT) toxicity (i.e.: Li-Fraumeni, ATM deficiency, active scleroderma, etc.) * Participants with a prior diagnosis of cerebrovascular accident (CVA) or transient ischemic attack (TIA) * Pregnant or breastfeeding (NOTE: breast milk cannot be stored for future use while the mother is being treated on study) * Known additional malignancy that is active and/or progressive requiring treatment; exceptions include basal cell or squamous cell skin cancer, in situ cervical or bladder cancer, or other cancer for which the subject has been disease-free for at least three years prior to enrollment * Prolonged Fridericia's corrected QT interval (QTcF) \> 450 ms for men and \> 470 ms for women at time of enrollment * Subjects with symptoms of ischemic cardiac disease, congestive heart failure, or myocardial infarction within 6 months of registration and/or uncontrolled cardiac rhythm disturbance * Subjects with a pulmonary embolism, deep vein thrombosis, or prior clinically significant venous or non-CVA/TIA arterial thromboembolic event (e.g., internal jugular vein thrombosis) within 3 months prior to enrollment * Patients with a history of a venous or arterial thromboembolic event must be asymptomatic prior to enrollment and must be receiving a stable regimen of therapeutic anticoagulation (low molecular weight heparin (LMWH) or direct oral anticoagulation (DOAC)). Use of coumadin is permitted; however, therapeutic dosing should target a specific international normalized ratio (INR) stable for at least 4 weeks prior to enrollment. NKTR-214 has the potential to down-regulate metabolizing enzymes for coumadin for approximately 1 week after administration of each dose of NKTR-214. Due to the possibility of drug-drug interactions between coumadin and NKTR-214, frequent monitoring of INR and ongoing consideration of dose adjustments are warranted throughout the patient's participation on study. * Subjects with significant psychiatric disabilities or seizure disorders if considered unsafe in the opinion of the treating physician * Subjects with symptomatic pleural effusions or ascites * Subjects with organ allografts * Subjects who require, or are likely to require, systemic treatment doses of corticosteroids, or other immunosuppressive drugs, or have used them within 2 weeks of registration (clarification: subjects receiving physiologic maintenance or replacement doses of systemic steroids or inhaled steroids are eligible) * Subjects with known human immunodeficiency virus (HIV) infection, active or chronic hepatitis B or hepatitis C infection, or with clinical evidence of hepatitis 15. Subjects with known hypersensitivity to IL2 or those who experienced significant immune-related AEs requiring treatment with steroids or other immunosuppressant therapy during prior treatment with ipilimumab, or anti- PD-1/PD-L1 checkpoint blockade therapy 16. Subjects who cannot provide independent, legal, informed consent