Mayor Marion Barry, who declared in his second inaugural address in January that reducing unemployment was the "most urgent task" facing his administration, conceded yesterday that his programs at best would put "a small dent" in the District's troublesome 10.4 percent unemployment rate.

"I've never said that this administration would be able to solve the problem of joblessness," Barry said during his monthly press conference, where he also gave a status report on the city's summer youth employment program.

"What I have said is we'll make the effort to do all we could," he added. "It's going to take massive help from the federal government in terms of jobs training and subsidized jobs and public works."

Barry was criticized last month by City Council members Charlene Drew Jarvis (D-Ward 4) and H.R. Crawford (D-Ward 7) for cutting $1 million from this year's budget for year-round job training programs as part of his effort to avert a potential deficit.

Through a combination of federal and city budget cuts, the summer youth program has been scaled down from a high of 28,000 jobs in 1979, the year Barry first took office, to 19,000 jobs last summer and an estimated 17,000 this year, according to D.C. Department of Employment Services figures. The unemployment rate among minority youth in Washington has been hovering around 42 percent for the past several months.

Bernard Demczuk, an organizer and lobbyist for the American Federation of Government Employees (AFGE) and one of the mayor's staunchest backers among organized labor, yesterday criticized Barry for cutting back job-training programs.

"I'm very, very displeased with Barry," Demczuk said. "Jobs and more jobs should be the priority of every administration . . . . I'm in sympathy with the mayor's predicament in the 1983-1984 budget cycle, but no monies should be taken from jobs-training programs. He can't back off."

At his press conference, Barry said the cuts he ordered in job-training programs for high school dropouts and adults with dependents were needed because of a tight budget and a "fragile" economy.

"I make no apologies for the fact that because of our budget difficulties, we had to make some slight reductions," Barry said. "But those reductions have been minor compared to cuts in programs in other cities. We can't put our heads in the sand and act as though we don't have a recession going on."

Barry said he was "just elated" by this year's summer jobs program, despite the reduction in the number of openings. The program once offered eight weeks of employment, but since last summer the city has reduced that to seven weeks. The city found that the shortened program led to a lower dropout rate and enabled the city to give jobs to more young people, according to Matthew Shannon, director of the Department of Employment Services.

The city this year has placed a total of 18,456 people in the program to start, anticipating that between 10 and 15 percent will not show up. About 8,800 of the total jobs filled will be funded by the federal government and 7,500 will be paid for by the District. An additional 1,000 jobs will be offered by the private sector.

The program this year targeted and gave preference to particular disadvantaged groups: high-school dropouts, single parents, criminal offenders, handicapped individuals and Hispanics. About 3,260 of the participants are in one of these categories, though Shannon said the number of Hispanic applicants was "discouraging."

Barry said this targeting marks the start of an effort to encourage youths who rely on city assistance, including 19-to-21-year-olds in the foster care system, to become more financially independent.