Health Career Advancement and Remuneration Exclusion for Training Act or the Health CARE Training Act This bill adds requirements to and otherwise modifies the Health Profession Opportunity Grants program. The program awards grants for demonstration projects to provide Temporary Assistance for Needy Families recipients and other low-income individuals with education and training for health care occupations that pay well and have labor shortages or are in high demand. Specifically, entities administering the projects must provide participants with education and training to obtain a post-secondary credential that meets the standards of the state in which the project is conducted or that meets standards set by the Department of Health and Human Services, as applicable. In addition, the bill expressly excludes cash stipends or emergency assistance paid to project participants from income for federal income tax purposes.