The time between the end of the school day and the traditional end of the workday can be a logistical nightmare for working parents in cities like New York.

One candidate in the mayor’s race has a far-reaching plan to help ease the high-stakes scramble to secure child care or paid activities to keep students busy after classes end: a universal after-school program.

Zellnor Myrie, a Democratic state senator from Brooklyn, on Thursday is expected to propose offering an after-school seat to every public school student who wants one. The sweeping program would be free and cover all children — from 3-year-olds in preschool to high school seniors — and would run until 6 p.m. on weekdays.

Mr. Myrie’s plan would also guarantee spots in the city’s no-cost summer academic and enrichment youth program. Tens of thousands of children typically languish on wait-lists or are denied seats every year.

The proposals would represent a huge undertaking in the nation’s largest school district, where most preschool classes run only until the school day ends, and where hundreds of public schools lack free, city-funded services after dismissal. Mr. Myrie did not offer a detailed funding plan but said he would pull from several sources, including savings from reduced local spending on migrants and revenue collected from audits of unpaid taxes owed to the city.

More than 20 million U.S. children are unable to sign up for an after-school program, despite interest from their families, national surveys show. Many parents struggle to find open seats, while others simply cannot afford to spend several hundred dollars a month on programs. Lower-income families often have the toughest time finding high-quality options.