This phase I trial studies the side effects and best dose of berzosertib (M6620) when given together with cisplatin and radiation therapy in treating patients with head and neck squamous cell carcinoma that has spread from where it started to nearby tissue or lymph nodes (locally advanced). ATR kinase inhibitor M6620 may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. Drugs used in chemotherapy, such as cisplatin, 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. Radiation therapy uses high energy x-rays to kill tumor cells and shrink tumors. Giving ATR kinase inhibitor M6620 together with cisplatin and radiation therapy may work better in treating patients with locally advanced head and neck squamous cell carcinoma.