Healing Communities through Health Care Act This bill establishes and revises programs to support health care and related services in communities that exhibit indicators that lead to adverse health outcomes. Specifically, the bill requires the Department of Health and Human Services (HHS) to conduct innovative demonstration projects through the Medicaid program in six states to provide housing services to individuals with behavioral health needs who are experiencing housing insecurity, including referrals to housing services from within a health care setting. The bill reauthorizes the HHS health-care occupation education and training grants program for low-income individuals and extends eligibility for such grants to hospitals. The National Institutes of Health also must establish and support a clinical research network to study trauma-informed, community-based interventions for victims of violence. Further, the Health Resources and Services Administration (HRSA) must grant a preference for funding specified health career programs in communities that have high rates of community trauma, such as elevated intergenerational poverty, drug-overdose deaths, or violence-related injuries or deaths. The bill also reauthorizes through FY2025 the HRSA National Health Service Corps, which supports education and student loan repayment for eligible primary care providers in health-professional shortage areas. Additionally, the bill expands the availability of funds for local workforce boards to support employer-sponsored employee training programs that have demonstrated specified results.