Clayton Yeutter arrived in steamy Washington during the summer to fill one of the hottest of the hot seats in the Reagan administration, becoming the U.S. Trade Representative at a time when trade was becoming one of the White House's top priorities.

Since he was sworn in July 1, Yeutter has been thrust into the limelight far more than his better-known predecessor, Labor Secretary William E. Brock, was during his early days on the job. Yeutter's only news conferences since taking over have been at the White House -- two in the past two weeks, with another likely later this month as the president refines his new, more aggressive stance on trade. And the pile of newspaper clippings on trade collected by the office is twice as big as it was a few months ago.

Yeutter's free-trade orientation and his self-effacing public affirmation that he speaks for the president instead of himself seem likely to endear him to the powers that run the White House. "Doesn't really matter whether I think it's a good idea," he said at one White House news briefing. "It depends on whether or not the president thinks it is a good idea."

He has, moreover, worked hard during the past month and a half to smooth relations between his office and the Commerce Department, which had become ragged during the past two years. Yeutter got together early in the summer with Commerce Secretary Malcolm Baldrige to try to make sure the two agencies can work together. The contacts have continued with the confirmation of Bruce Smart as the undersecretary of Commerce for international trade.

As a result, both agencies were reported to have had a major role in shaping the president's new trade policy.

The relationship had become strained because of Baldrige's push for a separate trade department, which, in effect, would have swallowed the USTR trade specialists into a larger agency. Since the president decided against pursuing that objective this year, a top trade official said, Commerce and the trade representative's office had to work more closely. YOU SAY YUTTER, I SAY YOITER . . . The pronunciation of Yeutter's name has caused great consternation among Japanese newsmen, who cover U.S. trade affairs very closely and have to transliterate his name phonetically into Japanese characters for their papers. For the record, Yeutter's office says his name is pronounced with a long "i," as in "writer." NEW FACES . . . Yeutter has appointed new people to the top jobs of his small agency and reshuffled key assignments of career trade specialists.

M. Alan Woods has been nominated to be the second deputy U.S. trade representative in Washington, and Alan B. Holmer has come from the Commerce Department to be general counsel. Charles R. Carlisle has arrived from the State Department to become chief textile negotiator, and Roger Bolton has transferred from deputy assistant Treasury secretary to be director of public affairs and public liaison.

James W. Frierson, a holdover from Brock's staff, will be chief of staff. Susan Nelson, who has worked with Yeutter for some 20 years, will be the trade representative's assistant.

Woods, who will succeed Robert E. Lighthizer, served as assistant secretary of defense and deputy director of presidential personnel in the Ford administration. He also has been a vice president of Sears World Trade and vice president of the Washington consulting firm DGA International.

Holmer had been deputy assistant secretary of Commerce for import administration and served in the Reagan White House as deputy assistant to the president for intergovernmental affairs. Carlisle was a special negotiator with the rank of ambassador in the State Department's Bureau of Economic Affairs.

Bolton was director of speechwriting for the '84 Reagan-Bush campaign and press secretary to the congressional Joint Economic Committee. Desiree Tucker will remain in the public affairs office.

Michael B. Smith will remain as deputy U.S. Trade Representative in charge of trade policy and negotiations. Woods will be responsible for management, congressional relations, public affairs and trade issues involving large sectors of the economy.

Yeutter has not filled one key position -- assistant U.S. Trade Representative for Japan. James M. Murphy Jr., a Japan specialist who had held the job, has become assistant U.S. trade representative for Europe and the Mediterranean.

Alex Platt, who had been the acting assistant U.S. Trade Representative for congressional affairs, will move to the National Security Council to specialize in trade.