WASHINGTON—The government is preparing to send one-time payments to most Americans to help them cope with the coronavirus outbreak, but that is little comfort for many college students and adult dependents who are left out.

The economic-relief law signed by President Trump on Friday provides $1,200 to most adults and $500 for children under age 17. That money—$292 billion—will start flowing within weeks from the Internal Revenue Service into bank accounts. People with little or no income can qualify, which means money will...