Housing Waiver Flexibility Act of 2020 This bill addresses the administration and funding of certain housing assistance programs in relation to the COVID-19 (i.e., coronavirus disease 2019) public health emergency. The Department of Housing and Urban Development (HUD) may waive requirements or specify alternative requirements related to the use of funds from the Housing Trust Fund and the self-help housing assistance program, except for requirements related to fair housing, nondiscrimination, labor standards, and the environment, if doing so is necessary to respond to the COVID-19 emergency. (The Housing Trust Fund provides grants to states for housing for low-income families, while the self-help housing assistance program provides grants to organizations that facilitate home ownership.) Furthermore, funds previously made available for the Choice Neighborhoods Initiative, a program that supports local efforts to revitalize public and HUD-assisted housing, shall remain available through FY2021. Additionally, the bill invalidates the FY2020 request for applications for funding under the Continuum of Care program, which funds efforts to address homelessness. Instead, HUD shall distribute amounts for FY2020 based on the results of the competition for FY2019.