UNITED NATIONS — Secretary-General Ban wants Congress to increase funding for U.N. peacekeeping by $200 million a year. Washington wants more American-friendly U.N. involvement in Iraq, fairer treatment of Israel at Turtle Bay, and a show of some backbone in confronting what President Bush has called “extremism.”

Returning yesterday to Turtle Bay from a two-day trip to Washington, Mr. Ban said it was time “to look for better days” in a relationship between America and the United Nations that at times “was not easy.”

At the United Nations, where there is never a shortage of hostility toward America, some were outraged that while visiting the White House on Tuesday, Mr. Ban called Mr. Bush a “great leader.” Mr. Ban felt compelled to explain to reporters yesterday that “in diplomacy, it is appropriate to address heads of state or government with due respect and courtesy.”

Without America, “neither of us would be here,” the new chairman of the House Committee on Foreign Affairs, Rep. Tom Lantos, a Democrat of California, reminded his South Korean guest during a Washington reception Tuesday night.

A Holocaust survivor, Mr. Lantos said he would have been killed by the Nazis, and Mr. Ban would have become a “nameless, faceless inhabitant of a united North Korea” if America had not interfered.

In an hour-long White House meeting — longer than planned — the new secretary-general talked with his host about how they might help each other. According to one participant, Mr. Ban stressed the need for more diplomatic involvement with Iraq’s neighbors — a different approach from Mr. Bush’s new plan. Nevertheless, the conversation showed more agreements than differences, the source said.

American officials who follow Turtle Bay closely have recently started to complain about the slow pace of Mr. Ban’s decision-making as he goes on restructuring the organization. They fear that a lengthened process could diminish Mr. Ban’s ability to make tough changes.

“The jury is out,” one American official said yesterday, assessing Mr. Ban’s performance on the condition of anonymity. “The question is, ‘Why is the jury still out?'” Mr. Ban has already named a few new top officials, most of them from poorer countries. He now plans to name an American, Lynn Pascoe, to head the influential political department, which may devour a U.N. unit charged with disarmament as well as a new department created last year by Mr. Ban’s predecessor, Kofi Annan, for “peace building.”

Fearing for their jobs, some officials at those two departments are lobbying against the growing American influence — a sentiment that always has a constituency at Turtle Bay. Another restructuring attempt that is favored by Washington — splitting the peacekeeping department to a support and operational units — has also met with increasing resistance from Third World countries that can block U.N. budgets.

“Please, wait a little bit until I finish,” Mr. Ban said yesterday, adding he intends to “finish, if possible, all the appointments at one time.”

At a Washington meeting with congressional foreign-policy and budgetary leaders, meanwhile, Mr. Ban asked to amend a legislation that allows Congress to approve payments of no more than 25% of the U.N. peacekeeping operations, while the United Nations assesses American dues at 27%. America pays over $1 billion for U.N. peacekeeping annually.

“There is a shortage of two percentage points, which will result in annually a $150 million or $200 million shortage of American contributions, which will, if it is accumulated, create very difficult constraints in smoothly carrying out peacekeeping operations,” he told reporters yesterday.

However, from the White House’s point of view, it was unfortunate that on the day Mr. Bush hosted the new U.N. chief, newspaper headlines referred to a Turtle Bay report of 34,000 Iraqis deaths last year, which allowed the New York Times to refer to the U.N. body count as “a vivid measure of failure” of the Iraqi government and the Bush administration.

Americans are unhappy with the United Nations’s performance in Iraq, arguing it could do more. One change considered at Turtle Bay is replacing the whole Iraq team, which has been headed since the summer of 2004 by a Pakistani diplomat, Ashraf Qazi. Mr. Qazi’s deputy, Michael von der Schulenburg of Germany, who is a longtime U.N. employee, previously held posts in Iran and Syria, two neighbors that Washington considers unhelpful in Iraq.

At a meeting with congressional leaders yesterday, several participants of both parties complained to Mr. Ban about an anti-Israel bias at the General Assembly and the Human Rights Council and about large budgets for endeavors such as the “Committee on the Exercise of the Inalienable Rights of the Palestinian People.”

Separately, Mr. Ban yesterday called on diplomats to “abide by and comply with all necessary regulations in force in the country where one is working.” He answered a question about the accumulation by Turtle Bay envoys of $18 million in unpaid New York parking tickets.