Mothers and Newborns Success Act This bill directs the Health Resources Services Administration (HRSA), the Centers for Disease Control and Prevention (CDC), and the National Institutes of Health (NIH) to take certain actions to improve, and reduce racial, ethnic, and geographic disparities in, maternal health outcomes. First, with respect to HRSA, the bill provides statutory authority for grants that support (1) state efforts to reduce disparities in maternal health; and (2) increased access to maternal and obstetrics care in rural areas, including through telehealth. HRSA must also award grants to identify best practices to improve pregnancy and postpartum outcomes. Second, the bill authorizes the CDC to establish or expand, and provide technical assistance on capturing racial and other demographic data related to, maternal health data collection. It also expands the scope of safe motherhood grants awarded to government health departments to include (1) classifying birthing facilities based on the level of maternal and neonatal care the facilities provide, and (2) improving the validity of pregnancy data on death certificates. In addition, the CDC must award grants for a campaign targeting health care providers and the public to increase awareness of maternal and child health issues. Lastly, the NIH must establish the National Maternal Health Research Network to support efforts to improve diagnostics and clinical treatments, improve health outcomes, and reduce inequities to reduce maternal mortality and promote maternal health.