Promoting Resources to Expand Vaccination, Education, and New Treatments for HPV Cancers Act of 2021 or the PREVENT HPV Cancers Act of 2021 This bill reauthorizes through FY2026 and expands Department of Health and Human Services activities to educate health care providers and the public about human papillomavirus (HPV), its association with certain cancers, and the importance of HPV vaccines. Specifically, the bill (1) requires relevant educational materials to include information about both cervical cancer and other HPV-associated cancers and about the importance of HPV vaccines, including for males; (2) expands the scope of a national education campaign about gynecologic cancers to address HPV-associated cancers more broadly and target specified, at-risk populations; and (3) authorizes demonstration projects to increase knowledge and awareness of gynecologic cancers.