Mentoring and Supporting Families Act This bill establishes demonstration projects for peer support, mentoring, and career coaching under the Health Profession Opportunity Grants program, which supports organizations that provide low-income individuals with education and training for health care occupations. Specifically, the Administration for Children and Families (ACF) must award grants to organizations that provide case management plans to help participants develop workforce skills on an ongoing basis. The ACF must give preference to project applications that provide participants with a cash stipend.