But Mr. Hayes said that while a strong majority of Republican voters adored Mr. Trump, there are creeping doubts about other administration advisers. Mr. Hayes said that Mr. Scaramucci’s interview with The New Yorker magazine, in which he savaged several White House colleagues in sexually graphic terms, had shocked Republicans in his state.
“How does that help us get health care and tax reform and rebuilding the military?” Mr. Hayes said.
Gov. John Kasich of Ohio, a Republican who has been critical of Mr. Trump, echoed that sentiment, saying meaningful policies will emerge from the White House only when the “chaos” in the administration abates. He said he was uncertain whether the shake-up of the senior staff would have that effect.
“You’ll have optimism within the White House when they start having stability,” Mr. Kasich said.
Among the president’s legislative allies in Washington, too, there is a deepening sense of dread that presidential tweets — like the out-of-the-blue ban on transgender people serving in the military — and continuing chaos inside the West Wing will get in the way of efforts to lower taxes, crack down on immigration, overhaul trade policies and rethink the country’s foreign policy.
“The administration is having a hard time getting out of its own way,” said Ira Mehlman, a spokesman for the Federation for American Immigration Reform, which was angry about the president’s criticism of Mr. Sessions. “The seeming disarray in the White House obviously makes it hard for the administration to carry out its policies.”
Joseph A. Trillo, a former Republican National Committee member from Rhode Island who was chairman of Mr. Trump’s campaign in the state, faulted others for the White House disarray and chalked up any missteps to Mr. Trump’s newcomer status in Washington.
“He’s made some mistakes,” Mr. Trillo said of Mr. Trump. “He didn’t have political experience, and I think some of the biggest mistakes are some of the people he has surrounded himself with.”
The turbulent phase appears to have taken its toll on Mr. Trump’s popularity, even among those in his own party. Though Republicans are strongly supportive of him over all, public polls have shown dissatisfaction on the right with his personal demeanor and Twitter habits. On Friday, a Gallup tracking poll found Mr. Trump’s job approval rating was 39 percent.
Inside Washington, the reservations run even deeper. Some veteran Republican lobbyists are increasingly skeptical that the president has built a team capable of making good on his promises. At the end of a week in which the party failed in its promise to repeal former President Barack Obama’s health care law, one Washington lobbyist, who did not want to be identified as being critical of the president, said he and others were frustrated, appalled and scared.
The concern, the lobbyist said, is that without sustained White House leadership — the kind that is in short order — complicated legislation like a tax overhaul or rolling back banking regulations will not be accomplished.
Andrew Roth, the chief lobbyist for the Club for Growth, a group that fiercely advocates lower taxes, expressed optimism that Republicans would succeed, and he said that some of Mr. Trump’s economic advisers were working effectively in spite of the chaos.
But Mr. Roth acknowledged that two things could get in the way: “Distractions being caused by a White House that is still in a transitional phase” and a “dysfunctional Republican Party” in Congress that includes too many liberals.
“It is well past time that people recognize that there are far too many Democrats in the Republican Party,” Mr. Roth said.
Republicans are hardly despondent across the board about the seemingly listless pace of change in Washington. In addition to the confirmation of a new Supreme Court justice, Neil M. Gorsuch, they take heart from the list of business regulations Mr. Trump has voided, and from his administration’s aggressive enforcement of immigration laws.
Jay Timmons, the president and chief executive of the National Association of Manufacturers, a powerful business lobby, said he was optimistic Republicans would enact major legislation around taxes, infrastructure investment and more.
“There’s a lot going on that has been beneficial to the business community,” Mr. Timmons said, acknowledging that there had also been distractions from the party’s main agenda. “That doesn’t mean that progress and success is still not occurring.”
Still, Republican activists and party officials described the Senate health care vote, held in the early hours of Friday morning, as a bitter disappointment, and several spoke in caustic language about the three Republican lawmakers who blocked the bill — Susan Collins of Maine, John McCain of Arizona and Lisa Murkowski of Alaska.
The sharpest frustration, however, came from Trump loyalists who described Congress as having failed to accommodate Mr. Trump and his agenda more broadly — or even as taking a deliberately oppositional posture toward a president of their own party.
“I blame everything on Congress, and most of the people I talk to feel the same way,” said Rex Early, an Indiana businessman who led Mr. Trump’s campaign in the state. “I’d like to see him take on Congress, but I think he feels that he has to get along with them, and he’s probably right.”
Mr. Trump has occasionally berated Republican members of Congress, and on Wednesday rebuked Ms. Murkowski on Twitter for having “let the Republicans, and our country, down” with her position on health care. On Friday, Mr. Trump stopped short of criticizing the health care holdouts by name, but lamented in a speech on Long Island that “the swamp” had prevailed over his agenda, for now.
But some of his supporters would like to see Mr. Trump go further, and a number of activists and Republican candidates called on Mr. Trump to take a harder line with members of his own party.
Corey Stewart, a conservative immigration activist in Virginia who nearly captured the party’s nomination for governor this year, encouraged Mr. Trump to take the fight more aggressively to intransigent Republicans.
“He’s been remarkably patient,” said Mr. Stewart, who has announced that he will run for Senate in 2018. “I think he needs to play a little bit more rough with the Republican establishment in the House and Senate.”
Mr. Stewart, too, said that there were aspects of the president’s conduct that appeared unproductive, like his public feud with Mr. Sessions.
“This stuff would be better solved behind closed doors,” Mr. Stewart said.
But other supporters said that after six months with Mr. Trump in office, they do not expect a change in his behavior — and many do not want one.
Pam Bondi, the attorney general of Florida and a strong Trump supporter, suggested the onus was on Congress to catch up with Mr. Trump. “President Trump is ready and waiting for them to act,” Ms. Bondi said, referring to the health care issue.
“Congress should beware,” she added, “our president will not give up on doing what’s right for the American people.”