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). Radiation therapy uses high energy radiation or protons to kill tumor cells and shrink tumors. Chemotherapy drugs, such as cisplatin and carboplatin, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. 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 head and neck squamous cell carcinoma.