Nothing bugs David Clarke more than the suggestion that Marion Barry helped him win his new job as City Council chairman and that Clarke is beholden to the mayor.

Clarke acknowledges a small debt to Barry for slipping him helpful polling data late in his uphill campaign for the Democratic nomination last fall. But Clarke credits his stunning victory over incumbent Arrington Dixon and former council chairman Sterling Tucker to his own vigorous campaigning and the nearly unanimous support of organized labor.

Since taking office three weeks ago, Clarke has moved swiftly to put his personal imprint on the chairman's office and to signal both to Barry and to his own friends in organized labor that he intends to chart an independent course.

For openers, Clarke greeted Barry's 1984 budget proposal with stinging criticism, saying that the mayor was more interested in keeping his campaign pledge of pay increases and no layoffs for city employes than in protecting the poor from deep cuts in social services.

Clarke, a one-time civil rights activist and a former council member from Ward 1, appears ready to fight Barry's plan to eliminate the General Public Assistance Program and to cut spending for other social services. He also signaled at a budget hearing last week a willingness to spend more for public education next year than the mayor has proposed.

Clarke also surprised his stalwart labor allies by privately discussing with the mayor the possibility of reopening wage agreements with city employes to try to scale back a scheduled 7 percent pay increase set for Oct. 1. When labor leaders found out about the meeting and raised a fuss, Barry backed down but Clarke didn't.

Clarke's independent streak and his propensity for asking the toughest questions about Barry's budget proposal have irritated some of the mayor's aides but impressed others. "You better come in with your facts and figures," said one high-ranking administration official in describing the third degree that Clarke gives Barry's department heads.

Clarke, who often was given to brooding and emotional outbursts, seems to have gained considerable poise and confidence since becoming chairman.

He reorganized the council's committee system with little of the rancor that marked the last reorganization two years ago. He deftly shifted authority to amend the mayor's budget proposals from individual committees to the council's committee of the whole. As chairman of that committee, Clarke will have a powerful base from which to offer alternatives to the mayor's spending plans.

Finally, he installed his own people in most key council staff positions, but wisely retained one of Dixon's most knowledgeable aides, legislative counsel Bruce C. French, to help in the transition.

But there have been rough spots. Clarke still occasionally displays traces of his quick temper and haughtiness--traits that marred his tenure as a council member and caused some to wonder whether he would be able to effectively lead the often quarrelsome council.

At last week's council session, Clarke impatiently schooled a colleague on "point of order" parliamentary procedures. He grimaced and groaned noticeably at a budget hearing when one of the mayor's department heads couldn't answer some of his questions about the city's housing programs.

Most District Building observers seem to agree that Clarke has gotten off to a strong start, but his ability to lead the council and assert his independence from the mayor will be put to a much tougher test in the next 30 days when debate over the new budget heats up and the council, including quite a few Barry partisans, must act.

As Clarke well understands, it's one thing to ask tough questions about Barry's budget. It's quite another to muster the votes to change it.