This phase II trial tests the addition of chemotherapy, with carboplatin and paclitaxel, or chemo-immunotherapy, with carboplatin, paclitaxel and cemiplimab to standard salvage surgery followed by post operative radiation therapy and cisplatin for high risk patients, for the treatment of patients with PD-L1 positive head and neck squamous cell carcinoma that has come back and spread to nearby tissue or lymph nodes after a period of improvement (locally recurrent) or is persistent. Carboplatin is in a class of medications known as platinum-containing compounds. It works in a way similar to the anticancer drug cisplatin, but may be better tolerated than cisplatin. Carboplatin works by killing, stopping or slowing the growth of cancer cells. Paclitaxel is in a class of medications called antimicrotubule agents. It stops cancer cells from growing and dividing and may kill them. Immunotherapy with monoclonal antibodies, such as cemiplimab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Salvage surgery is surgery that takes place to remove tumor tissue after a failure of other treatment. High risk patients also receive radiation therapy uses high energy x-rays, particles, or radioactive seeds to kill cancer cells and shrink tumors. Cisplatin is in a class of medications known as platinum-containing compounds. It works by killing, stopping or slowing the growth of cancer cells. Adding chemotherapy or chemo-immunotherapy to standard salvage surgery may kill more tumor cells than salvage surgery alone in patients with PD-L1 positive locally recurrent or persistent head and neck squamous cell carcinoma.

This phase II trial studies the effect of pembrolizumab alone compared to the usual approach (chemotherapy \[cisplatin and carboplatin\] 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. Cisplatin is in a class of medications known as platinum-containing compounds. It works by killing, stopping or slowing the growth of cancer cells. Carboplatin is also in a class of medications known as platinum-containing compounds. It works in a way similar to the anticancer drug cisplatin, but may be better tolerated than cisplatin. Carboplatin works by killing, stopping or slowing the growth of cancer cells. 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 alone after surgery may work better than the usual approach in shrinking recurrent or primary head and neck squamous cell carcinoma.