This trial is testing whether adding a new drug, ipatasertib, to an existing treatment, pembrolizumab, can better treat head and neck cancer that has returned or spread. Ipatasertib may help stop cancer growth, and pembrolizumab helps the immune system attack the cancer. The goal is to see if this combination works better than using pembrolizumab alone. Pembrolizumab has been used in combination with chemotherapy for various cancers, including head and neck cancer, and has shown improved overall survival and response rates.