Urban Progress Act of 2020 This bill addresses economic development, employment opportunities, gun violence, and poverty. Among other provisions, the bill makes permanent the work opportunity tax credit and increases it for employers who hire ex-felons; establishes grants to promote commercial development in underserved communities, to provide career training programs, and to support summer employment opportunities for youth; increases Supplemental Nutrition Assistance Program benefits when school is not in session for households with minor children; increases the child tax credit for low-income taxpayers; broadens the categories of prohibited persons who are barred from receiving or possessing firearms; requires states to submit mental health and criminal history records to the National Instant Criminal Background Check System; prohibits firearm transfers between private parties unless a licensed dealer first conducts a background check; requires gun owners to report lost or stolen firearms; and prohibits licensed gun dealers from transferring firearms to unlicensed persons prior to background check completion.