Testing the Addition of Anti-Cancer Drug, Cetuximab, to Standard of Care Treatment (Pembrolizumab) for Returning or Spreading Head and Neck Cancer After Previous Treatment

Inclusion Criteria: * Histologically confirmed diagnosis head and neck squamous cell carcinomas (HNSCC). * Previously untreated for recurrent and/or metastatic disease incurable by local therapies. * Primary tumor location of oral cavity, oropharynx, larynx, or hypopharynx. * Note: Other primary tumor sites of HNSCC, including nasopharynx primary tumor are not eligible. Unknown primary tumors may be eligible and can be enrolled at the discretion of the treatment team with approval by the study chair. * Measurable disease. * Must have platinum-refractory disease defined as disease progression during or ≤ 29 weeks after completion of definitive therapy (chemoradiation therapy) or adjuvant (post-operative) therapy. * Patient must have a combined positive score PD-L1 positive (CPS \>/= 1) tumor. * Any radiation therapy must be completed \>= 10 days prior to registration. * Patients should not have received any prior treatment in the recurrent or metastatic setting. * Prior therapy with neoadjuvant or induction anti PD-1/PD-L1 monoclonal antibody or cetuximab in the curative setting is allowed if last treatment dose was \>= 26 weeks prior to registration without evidence of disease progression during that treatment period. * Patient has not received a live vaccine within 30 days prior to registration. * Patient does not have a history of any contraindication or has a severe hypersensitivity to any component of pembrolizumab or cetuximab (≥ grade 3). * Patient has not received …