If issues are to play a part in the Democratic mayoral primary, then Mayor Marion Barry appears vulnerable for his handling of public school problems and the city's relations with Congress, according to the findings of a Washington Post election campaign poll.

Although he leads Patricia Roberts Harris, his chief opponent, by 13 points in the poll, Barry is seen as less capable than Harris in coping with those issues. In addition, to a lesser degree, Barry appears vulnerable on the question of which candidate can best bring honest government to the city.

But the Democrats interviewed rate Barry better than Harris in curbing crime, maintaining city streets, attracting new jobs and stimulating commercial development.

The Post survey of 1,374 registered Democrats also indicates that Harris, a Carter administration cabinet officer, has made little headway in persuading voters she can make the city run more efficiently, a major theme of her campaign.

Democrats were nearly unanimous in agreeing that a mayor can see to it that the city is run efficiently, but only 27 percent of them said Harris would be best suited for that task, compared to 33 percent for Barry.

Sharon Pratt Dixon, Harris's campaign director, said Friday that the polling results for the most part reflect Barry's election-year efforts to exploit the incumbency to improve his image.

Dixon said the mayor has received considerable publicity for his recent drive to repair and clean up streets, beef up police patrols and offer summer jobs to young people. But as the campaign wears on and Harris becomes better known, voters will realize that government services can be vastly improved, she said.

"There's no question we have to get Pat's message out to the people of the District of Columbia," Dixon said. "What Marion has done is more inculcated in the minds of the public than what Pat has done. The sense of what she would be like as mayor hasn't gotten out yet."

Peter Hart, Harris' pollster, said it might be too soon to ask voters to make meaningful comparisons of Barry's and Harris' administrative skills.

"What you're getting, as much as anything, is an evaluation of Barry, rather than a comparison," Hart said. "I don't think we're really getting a rating of Harris at this stage of the game."

Hart said his own polling revealed a widespread suspicion that "things are not very together" in D.C. government.

Barry and his campaign aides declined to comment on the Post poll, which was conducted June 16 to June 22 and shows Barry leading Harris by a margin of 45 percent to 32 percent with 10 weeks remaining before the primary.

The Post poll tried to determine what might be voting issues this fall by asking registered Democrats whether they felt a mayor could or could not "make much difference" in 12 specific problem areas. For each area in which citizens felt a mayor could make much difference, those interviewed were then asked which candidate they thought would be best at coping with that particular problem.

The areas included improving public schools, bringing jobs to the city, keeping down local taxes, curbing crime, maintaining good race relations, improving streets, providing honest government, keeping a lid on rents, limiting development in residential neighborhoods, providing efficient and responsive city services and stimulating downtown growth.

Voters appear to see little difference between Barry and Harris on the issues of taxes, race relations, rents, and development in residential areas. However, Harris appeared to hold a slight edge over Barry on the question of honesty in government.

Nearly nine out of 10 people interviewed said they thought a mayor can bring honest government to the District, but only 28 percent of them said they thought Barry was the one to do it. In fact, many of his own supporters failed to choose him as being the candidate best able to see to it that the government is run honestly.

Another 28 percent of the Democrats interviewed said Harris would be the best candidate to bring honest government to the District. But her showing was more impressive because she has a smaller political base than Barry.

Harris's strongest issue at this time appears to be education. Her pledge to provide leadership in that area and to end the mayor's adversarial relationship with the D.C. Board of Education over budget issues appears to be paying off.

About 87 percent of the Democrats interviewed said they believe a mayor can help to improve public schools, even though educational policy is set by an independent school board. Of those Democrats, 31 percent said Harris would be best able to improve the schools and 25 percent said Barry would.

Harris may be capitalizing on this issue most in Ward 3, an affluent area west of Rock Creek Park, one of the areas in the city where concern about the quality of public schools runs high. Barry, who narrowly won the 1978 Democratic mayoral primary with the support of white voters in Ward 3, is trailing Harris this year in that area, 38 pecent to 42 percent. Barry leads Harris in the city's seven other wards.

When it comes to relations with Congress, 88 percent of the Democrats interviewed believe a mayor can make a difference. Of those, 37 percent said Harris would do the best job of getting real help for the city from the Hill, compared to 28 percent for Barry.

Barry contends that relations between the city and Congress have never been better. As examples, he cites the record-high federal payment the city now is receiving and the fact that the vast majority of the legislation enacted by the City Council is approved by Congress (although late last year the House bowed to pressure from the Moral Majority and struck down a major revision in the city's criminal sex code).

For his handling of crime, street repairs, unemployment and downtown commercial development, Barry appears to get relatively high marks from Democrats interviewed.

Eighty-six percent of them said a mayor can make a difference in curbing crime. Of that group, 37 percent said Barry would do the best job and 20 percent said Harris would.

About 95 percent of the respondents said a mayor can improve streets and roads. Of those people, 35 percent said Barry would do the best job and 22 percent chose Harris.

As for bringing new jobs to the city, 89 percent of the citizens interviewed said a mayor can make a difference. Thirty-five percent of them said Barry would be best able to bring jobs to the city, compared to 26 percent for Harris.

Also, 93 percent of those interviewed said a mayor can help to stimulate downtown growth. Of that group, 38 percent said Barry would be best for that task, while 25 percent picked Harris.

City Council members John Ray and Charlene Drew Jarvis, two other candidates in the Democratic primary race, so far have failed to attract much popular support or recognition for being able to cope with city problems, according to the poll.

No more than 4 percent of the Democrats polled said they thought Ray was best suited to cope with any of the 12 issues listed. Five percent of those interviewed said they support or lean to Ray in the election.

As for Jarvis, no more than 3 percent of the Democrats interviewed said they thought she would best be able to cope with those problems--the same percentage that said they supported Jarvis for election.