This phase III trial compares cetuxumab to chemotherapy, carboplatin and paclitaxel, with intensity modulated radiation therapy for the treatment of patients with head and neck cancer who are unable to receive cisplatin. Cetuximab is in a class of medications called monoclonal antibodies. It binds to a protein called EGFR, which is found on some types of cancer cells. This may help keep cancer cells from growing. 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. Intensity modulated radiation therapy is a type of 3-dimensional radiation therapy that uses computer-generated images to show the size and shape of the tumor. Thin beams of radiation of different intensities are aimed at the tumor from many angles. This type of radiation therapy reduces the damage to healthy tissue near the tumor. It is not yet know if cetxiumab or chemotherapy, with intensity modulated radiation therapy works best for the treatment of patients with head and neck cancer who are unable to receive cisplatin.

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.