Mothers and Newborns Success Act This bill establishes and expands various health programs to improve, and reduce racial, ethnic, and geographic disparities in, maternal health outcomes. The Health Resources and Services Administration (HRSA) must establish a competitive grant program to identify best practices for improving maternal care and outcomes. The bill also requires HRSA to continue awarding grants for reducing disparities in maternal health and increasing access to maternal and obstetrics care in rural areas. The Centers for Disease Control and Prevention (CDC) must award grants for a campaign to educate health care providers and the public about maternal and child health issues. Additionally, the bill expands an initiative to promote safe motherhood by (1) requiring the CDC to award competitive grants for classifying birthing facilities based on the level of maternal and neonatal care that the facilities provide, and (2) allowing the CDC to award grants for improving the validity of pregnancy data that is captured on death certificates. The CDC may also establish or expand efforts to collect maternal health data, including by providing technical assistance to ensure that relevant data sets adequately represent communities of color. The National Institutes of Health must establish the National Maternal Health Research Network to support efforts to improve diagnostics, clinical treatments, and health outcomes; reduce inequities in maternal mortality; and promote maternal health.