President Reagan presented an $800,000 check to Mayor Marion Barry yesterday that will provide an additional 2,200 summer jobs to unemployed youths in the District.

Similar bonuses, from a $25.5 million discretionary fund, were given to other cities last month to put disadvantaged youths to work. The District was the last city to receive a bonus check, a Department of Labor spokeswoman said.

The youths to fill the additional jobs will be selected today through a "computer match-up" process and will begin work Monday, D.C. city officials said. Another 2,000 District youths who applied for jobs through the D.C. Department of Employment Services remain unemployed, they said.

A total of 18,456 youths between the ages of 14 and 21 are currently employed in the city's summer jobs program, according to officials. The youths earn the minimum wage of $3.35 and work four or five hours a day. The D.C. Department of Employment Services received $13.5 million in federal and local funds for the program.

At yesterday's White House ceremony, Reagan said that "when Labor Secretary Ray Donovan learned that my adopted hometown here was running out of money for its summer jobs program, he called Mayor Barry and offered to help. The result is today's check drawn from available funds at the Department of Labor."

Barry told the president that "this bonus . . . begins to put a dent in our large unemployment picture in the District of Columbia. Over 40 percent of our young people are out of work and obviously this won't solve the entire problem, but it goes a long way to begin to help it."

During the ceremony in the Rose Garden, Reagan again suggested that child-labor laws should be changed to permit youths under 16 to work in construction and other such jobs. Recalling that he got his first job with a construction company when he was 14 years old--in 1925--Reagan said, "I know that some regulations make that impossible today. Maybe we can keep on going 'till we change some of those back again."

Currently, construction work is barred to children under 16. Some 15- and 16-year-olds are permitted to work in offices, in retail stores and waiting on tables and washing dishes in restaurants, a Labor Department spokesman said.

Most of the youths for the additional jobs will be picked from a pool of 14- and 15-year-olds and will be placed in an academic-enrichment program called Associates for Renewal in Education Inc., D.C. officials said. The program, supervised by District teachers who are hired with summer job funds, provides youths in instruction in reading skills and use of a computer.