The city's entry into the race to build a civic center in the metropolitan area barely made it to the starting line last week, following some sharp political infighting between Mayor Walter E. Washington and the City Council.

At one point, during what was expected to be a routine briefing of the council by the major and city planning officials before the plans were made public, Council member Marion Barry walked out of the room in anger and had to be called back into the meeting by Council Chairman Sterling Tucker.

Council member Nadine Winter was dismayed by the lack of council input prior to the scheduled release of the civic center proposal Friday afternoon. She urged that no plans be made public for several days.

The mayor would not agree to that. But he did not acquiesce to council sensitivities by deciding not to read or make public the press release his staff had prepared to accompany the announcement because it implied council support of the proposed $110 million project at Mt. Vernon Square.

"The studies are a product of earlier consultations between Mayor Walter E. Washington and the D.C. Council, in cooperation with members of the business community and citizens representatives," the aborted news release read.

"Our studies have concluded," the mayor is quoted as being ready to say, "that the proposed civic center is economically feasible, that it will pay for itself within a period of 15 years, and that it will bring important dividends to this city in terms of economic activity, jobs and business development."

For city council members who had just been given the report moments earlier, that was too much - especially since the press release suggested that this was the official city government plan.

"It was politics, pure and simple politics," one supporter of the mayor said afterwards. To some members of the council, however, it was a showdown prompted by the mayor's apparent plans to leap into the limelight on an all-important project with only token input from the city's elected lawmakers.

For any possible mayoral candidate - and there were three of them in the room at the time - a key role in the civic center proposal and surrounding publicity was an attractive political plum.

The city's business community, which gives generous campaign contributions, is said to be strongly in favor of the center. The civic center package is also being touted as a way to bring jobs to the city and help stimulate the downtown economy.

The council must approve any legislation requestion authority to borrow money to finance center construction, and two of the key committee chairmen are considered possible opponents of the mayor in the 1978 election.

Candidate Barry is chairman of the finance and revenue committee, which could be assigned to consider part of the civic center proposal. Candidate Tucker, in addition to being chairman of the council, heads the budget committee.

Council member Winter is not a candidate for mayor, but she has been making considerable efforts on her own in favor of building a center and has already made it known that she opposes the design of the mayor's plan.

There have been some noteworthy ups and downs at the District Building since the end of the two-day takeover by Hanafi Muslims earlier this month.

Robert Woodley, the head of the building's security force, has been promoted from lieutenant to captain. James Yancy, the security guard who, according to Woodley, ran down the stairs when the first shots were fined during the takeover, has resigned, Woodley said. Yancy could not be reached for comment.

Mayor Washington, whose political stock was said to be slipping in pre-takeover days, gained points for the way the crisis was handled, his supporters say. That improves the mayor's potential for a federal appointment, but he might not take one if it were offered. The mayor seems to be thinking more and more these days about running for re-election in 1978.

It's not clear yet which - if any Democrats - will swithc labels by July 19 to run for the vacancy on the Cith Council created by Julius Hobson's death. "But," one D.C. Statehood Party member predicted last week, "watch how many hotshot Democrats suddenly become independents." Only non-Democrats can run in the special election.

One name being bandied about the District Building is that of Joe Yeldell, who Mayor Washington suspended as director of the Department of Human Resources on Dec. 3.

But count Yeldell out, his confidants say. He has already done his City Council bit and the salary - $26,000 a year - is a bit too low. Now, will he run for mayor?