Community Services Block Grant Enhancement Act of 2020 This bill modifies the distribution of COVID-19 (i.e., coronavirus disease 2019) relief funds with respect to, and temporarily expands eligibility criteria to participate in activities funded by, the Community Services Block Grant. (This grant provides formula-based allotments of federal funds to states, territories, and tribes for anti-poverty activities.) Specifically, the bill exempts funding appropriated for this grant under the CARES Act from a cap on the annual growth of a state's allotment and requires states to distribute COVID-19 relief funding to the local organizations and other eligible entities that carry out block grant activities within specified time frames. In addition, it expands eligibility for programs and activities funded by the block grant to 200% of the federal poverty line (FPL) through FY2022. Under current law, eligibility is set at the FPL, with states authorized to increase eligibility to 125% under certain conditions.