This randomized phase II trial studies how well Ataxia-telangiectasia and Rad3-related [ATR] kinase inhibitor VX-970 (VX-970) and gemcitabine hydrochloride work compared to standard treatment with gemcitabine hydrochloride alone in treating patients with ovarian, primary peritoneal, or fallopian tube cancer that has come back after a period of improvement (recurrent). ATR kinase inhibitor VX-970 may stop the growth of tumor cells by blocking an enzyme needed for cell growth, and may also help gemcitabine hydrochloride work better. Gemcitabine hydrochloride is a drug used in chemotherapy that works to stop the growth of tumor cells by blocking cells from growing and repairing themselves, causing them to die. It is not yet known whether adding ATR kinase inhibitor VX-970 to standard treatment with gemcitabine hydrochloride is more effective than gemcitabine hydrochloride alone in treating patients with ovarian, primary peritoneal, or fallopian tube cancer.