The Carter administration may soon face its first major credibility test with local political leaders in the battle over voting representation for the District in Congress. Just what role the White House plans to play in that struggle is one of the best-kept secrets in Washington.

In February, when his presidency was less than a month old, Carter said that he was in favor of the city having voting rights in Congress. But he never spelled out what those voting rights should be or what he would do to help bring them about.

Meanwhile, local political and civic leaders have made full voting representation - two senators and two representative instead of the present non-voting delegate in the House - priority No. 1. What they want now is some indication of what the White House plans to do.

Although they generally praised the atmosphere surrounding this week's discussion of the subject by the special White House task force on D.C. problems, Capitol Hill supporters of D.C. voting rights in Congress expressed apparent surprise at some White House attitudes.

Rep. Charles C. Diggs Jr., D-Mich., chairman of the House District Committee, hinted that the discussion of the issue seemed to be clouded by a lack of information on the part of some of those involved.

Without specifically referring to the White House, Diggs said, "Here is an issue that's been debated, redebated and regurgitated over several decades. What you're fighting is sort of a ho-hum, here-we-go-again attitude."

Del. Walter E. Fauntroy (D-D.C.), was a little more specific in his criticism. "I had the feeling that some of the President's representatives who are new to the issue of full-voting representation raised some old questions," he said. He wouldn't name names.

Martha (Bunny) Mitchell, the presidential assistant who is coordinator of the task force, flatly refused to outline a White House strategy. That might come next month, she said. Mitchell said the administration does not have a recent nose count on where the votes are for full voting representation on the Hill, or if the White House would recommend a limited form of D.C. voting rights.

A House subcommittee will begin hearings July 25 on a full voting rights bill for D.C. Last year, a House vote on a similar bill fell 21 votes short of the necessary two-thirds majority.

Local leaders are still optimistic, saying that the basic attitude of the administration plus changes in Congress make the forecast for victory much better than ever before. But the White House role could be crucial, they say, and what they want is a lot more than just a statement from the President.

Gary Myers, who was one of the lawyers for Joe Yeldell when the latter was jousting with Mayor Washington and the news media over Yeldell's job as Human Resources director, is running as a Republican for a Virginia legislative seat from Alexandria.

Mayor Washington's "town hall" meeting on Capital Hill last week proved to be a popular gathering place for some of those who always get mentioned as possible runners for something. They included former school superintendent Barbara Sizemore (now running for City Council), former Judge Harry T. Alexander (now running the Hanafi Muslims' defense) and former federal housing administrator H.R. Crawford (who insists he's not running for anything).

"The Volkswagen's in the shop." That was presidential assistant Martha (Bunny) Mitchell's explanation Tuesday for why, despite the talk of the Carter administration running a back-to-basics White House, she and colleague Jim Dyke were whisked away from a meeting at the District building in a big, black shiny Caprice.

Now if someone would only tell why Vice President Mondale has missed the last four meetings of the special White House task force on D.C. problems. He chaired the kickoff session of the panel in May and hasn't shown up since.