This phase IIa trial studies how well pan fibroblast growth factor receptor (FGFR) kinase inhibitor BGJ398 works in treating patients with FGFR1-3 translocated, mutated, or amplified head and neck cancer that has returned after a period of improvement. Pan FGFR kinase inhibitor BGJ398 may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth.