This randomized phase II trial studies how well cisplatin with or without WEE1 inhibitor MK-1775 works in treating patients with head and neck cancer that has come back or has spread to other parts of the body. Drugs used in chemotherapy, such as cisplatin, may prevent tumor cells from multiplying by damaging their deoxyribonucleic acid (DNA), which in turn stops the tumor from growing. WEE1 inhibitor MK-1775 may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. It is not yet known whether cisplatin is more effective with or without WEE1 inhibitor MK-1775 in treating patients with head and neck cancer.