This phase II trial studies the effect of pembrolizumab in combination with radiation therapy or pembrolizumab alone compared to the usual approach (chemotherapy plus radiation therapy) after surgery in treating patients with head and neck squamous cell carcinoma that has come back (recurrent) or patients with a second head and neck cancer that is not from metastasis (primary). Radiation therapy uses high energy radiation or protons to kill tumor cells and shrink tumors. Chemotherapy drugs, such as cisplatin and carboplatin kill tumor cells by stopping them from dividing. Immunotherapy with monoclonal antibodies, such as pembrolizumab, may help the body's immune system attack the cancer and may interfere with the ability of tumor cells to grow and spread. Giving pembrolizumab in combination with radiation therapy or pembrolizumab alone after surgery may work better than the usual approach in shrinking recurrent or primary head and neck squamous cell carcinoma.