This trial is testing a new drug called amivantamab alone and with two other drugs, pembrolizumab and paclitaxel. It targets patients with head and neck cancer that has come back or spread. Amivantamab blocks signals that help cancer grow, pembrolizumab helps the immune system fight cancer, and paclitaxel stops cancer cells from dividing. Pembrolizumab has shown benefits when used with chemotherapy.