Mayor Walter E. Washington called on members of the D.C City Council yesterday to stand firm and united with him in the face of increased allegations of mismanagement in city government that some believe are threatening the city's new self-government.

"You cannot, just in a climate of deceit and a climate of speculation and suspicion, permit that institution (city government) to be pulled apart by any individual or group of individuals," he told Council members and others at a breakfast at the Mayflower Hotel commemorating the second anniversity of District home rule.

"We all know that there are many people who never wanted us to have an inch of home rule, and they're still there and they're growing," he said.

The mayor termed the recent allegations "fractures" and compared them to the 1968 riots following the April 4 death of the Rev. Martin Luther King Jr. Like other fractures, he said, the most recent allegations can be overcome.

The mayor's 15-minute speech never made specific reference to newspaper articles of the last six weeks, which have reported allegations of gross mismangement in the city's largest agency, the Department of Human Resources, and most recently reported allegations of illegal cash payments during the mayor's own election campaign in 1974.

The remarks came at a time when, according to well-placed District Building sources, more and more of the mayor's time is being occupied dealing with the political crisis spawned by the allegations, and speculation is increasing that the mayor himself may not finish his remaining two years in office.

Last week City Council Chairman Sterling Tucker told an audience at the Washington Cathedral that he resented assertions that the continuing allegations rendered the city "unfit for home rule and incapable of a greater measure of self-government."

The mayor echoed some of those concerns yesterday, by laying part of the blame on forces beyond the city government's control, and also by suggesting that uneven standards were being used to judge the city's actions.

"I'm talking about a circumstance that's come upon us where we have all the responsibility and very limited authority, and people expect us to perform like we've got it all," the mayor said.

"You going to take the Congress back, they're having a little trouble," he joked, in an oblique reference to suggestions that home rule be taken from the city because of the recent allegations.

The mayor noted President Ford's proposal last week to grant statehood to Puerto Rico. Some political activists in the city have sought statehood for the District for some time. But rather than statehood now, the mayor said, he would favor full representation in Congress, full control of the city's budget and full taxing authority for the District government.

While many political observers have labeled the current crisis in city government the most serious since the establishment of home rule, Mayor Washington said yesterday that "the institution of this government and the institution of this city stands firm, firmer than it's ever been in the history of this nation."

"The institution of this government and of this city is viable and strong," he said. "The institution, you must remember, rises above the individual Council members, mayors, department heads and others."

During the past six weeks, the city government has been the subject of a constant stream of front-page newspaper stories alleging first that Joseph P. Yeldell, the suspended director of DHR, had abused the agency's hiring, leasing and contracting powers.

The accusations have broadened beyond Yeldell to include alleged questionable public conduct by the mayor's two top advisers - City Administrator Julian Dugas, a longtime friend, and Corporation Counsel John R. Risher Jr.

On Dec. 23, The Washington Post reported that $1,400 in secret cash payments allegedly were used to pay salaries of workers in the mayor's 1974 campaign, and were not reported as required by law.

The city's Board of Elections and Ethics is conducting a "preliminary inquiry" into the alleged campaign financing irregularities, and six investigations of Yeldell and DHR - including one by the U.S. attorney's office here - have been announced.