Mayor Marion Barry once complained publicly about the apparent incompetence of some of the city government's public relations specialists. For a month, Barry said, he tried in vain to get a well-paid aide to design a satisfactory poster for the summer-jobs-for-teenagers program.

A few days ago, a proud city aide finally showed the mayor a poster that met with Barry's approval. There was a picture of a smiling Marion Barry on top of the poster, a jagged yellow brick road with dollar signs stitched across the middle and a slogan borrowed from the hit soul musical "The Wiz" - "Ease on Down (the road) to a Summer Job."

The summer jobs poster, which is careful to note that this is "Mayor Marion S. Barry Jr.'s 1979" jobs program, is only one of several efforts the new mayor is making to dress up the image of his office and the often-criticized bureaucracy he heads. Among plans to improve the city's public information system, aides say, are the following:

Barry is overhauling "City News," the monthly 90,000-circulation tabloid launched by former Mayor Walter E. Washington. Under the new masthead of "City Hall New Times," the Barry administration hopes to chronicle the government's own news and views, especially those of city employes. The project is estimated to cost $30,000 to $35,000 a year.

Lisa Berg, a freelance photographer who worked in the mayor's campaign, has been retained on a $5,000-a-year contract to supplement the limited in-house photography operations of the D.C. government. Her job is to photograph the mayor at late-hour and weekend functions. Berg's pictures are sent out to groups who request them, or used in city brochures as substitutes for staid portraits of the mayor.

Not all is roses. The "open, competent and compassionate administration" advocated by Barry during his campaign has not yet fully arrived. Some of the mayor's press aides still cannot explain the particulars of some city department operations, and they sometimes forget the names of department heads. Top aides to the mayor often return reporters' telephone calls late or not at all.

Earlier this month, when reporters asked to accompany Barry on a trip to meet with inmates at the Lorton Correctional Facility, reporters were barred from the meeting. "This is not a show," explained Ivanhoe Donaldson, general assistant to the mayor.

With no reference to Donaldson, Alan F. Grip concedes that all is not perfect. "There are still some problems. I will admit that frankly," said Grip, director of the newly established Office of Communications. "But 95 percent of the time when somebody makes a public relations gaffe, I make sure it's not repeated. In certain situations, we're still in the learning process."

Grip's task is to oversee public relations operations for most departments of the administration except the mayor's office.

Florence Tate, Barry's personal press secretary and public relations adviser, said when the new regime took over Jan. 2, "Too few people knew what was going on, who was doing it, why it was being done and what the government was. It was just some nebulous kind of thing - the government."

In the closing years of Washington's 11-year administration, public relations in the District Building was sharply criticized. Neither the mayor nor many of his top aides talked openly with reporters on a regular basis. Many of Washington's closest advisers distrusted the news media, feeling it was out "to get" the mayor.

Grip sat in his office recently and pointed to one agency's annual report in which there was a reference to legislation passed by the City Council that had, in fact, never been passed. Another annual report featured a cover drawing resembling an etching from a fourth grade art class.

While many persons in the new administration - including the mayor himself - feel the news media is often too critical, too sensational or too "negative," distrust for the press in the Barry administration does not seem as deep-seated as in the Washington administration.

Grip said the new publication, scheduled to make its debut June 25, "is not just going to be a laundry list of the accomplishments of the mayor." News from the City Council and city courts, employe letters to, and answers from, the mayor and regular columns will be featured, Grip said.

About 70,000 copies of the publication will be printed. City employes will receive their newspapers along with their regular paychecks. Other copies will be distributed to church, community and neighborhood groups, according to Grip.

In the meantime, Grip is considering ways to improve the press relations practices of District Building department heads and their top aides. For the past two years, a survey of Washington reporters has listed the D.C. government as one of the least responsive agencies among local and federal jurisdictions in the area.

Grip would not say precisely what he would do to change those practices. But he was willing to predict that the District government would not be in the cellar when the responsive standings come out later this year. "I betcha," Grip said the other day, "that that doesn't happen this year."