COVID HCBS Relief Act of 2021 This bill temporarily increases the applicable Federal Medical Assistance Percentage (i.e., federal matching rate) under Medicaid for certain approved home- and community-based services that are provided between October 1, 2020, and September 30, 2022. As a condition for receiving the increased rate, a state must agree to undertake activities to improve the delivery of such services during and after the public health emergency relating to COVID-19 (i.e., coronavirus disease 2019), such as by providing additional benefits to home health workers and by helping individuals who were relocated to nursing facilities during the emergency move back to their homes.