Health Care Broadband Expansion During COVID-19 Act This bill temporarily expands the Rural Health Care Program of the Federal Communications Commission (FCC) in response to COVID-19 (i.e., coronavirus disease 2019). It also establishes and provides funding for the Emergency Rural Health Care Connectivity Fund through FY2022 to carry out the Rural Health Care Program. Currently, the FCC subsidizes 65% of the cost of broadband for eligible public and nonprofit rural health care facilities through the Rural Health Care Program. This bill expands the program to encompass health care providers not located in rural areas, including mobile and temporary facilities, and it increases the subsidy rate to 85%. The bill also requires the FCC to (1) promulgate regulations to expedite the receipt of funds under the program, and (2) release funding for all requests within a designated time frame.