White House officials struggled Monday to get beyond the latest leak to the news media that embarrassed and distracted the Trump administration, acknowledging that the ever-present fear of seeing their private conversations in the press or on TV had created “a very difficult work environment.”

The White House has been lambasted for five days after it was leaked that an aide said Sen. John McCain, who has brain cancer, didn’t matter because “he was dying anyway.”

President Trump has weathered an unprecedented amount of private or sensitive information oozing out of his administration, from details of his conversions inside the Oval Office to airing of the backbiting among top White House officials.

All of which has sparked questions about the loyalty of staff and his management style.

Mr. Trump called the leakers “traitors and cowards,” and he vowed to root them out. But he also balked at reports that characterize the White House as infected with leakers.

“The so-called leaks coming out of the White House are a massive over exaggeration put out by the Fake News Media in order to make us look as bad as possible,” he tweeted. “With that being said, leakers are traitors and cowards, and we will find out who they are!”

Mr. Trump has been plagued by leaks since his election. Obama administration officials during the transition and holdovers after the inauguration have been blamed for a flood of leaks about intercepted phone calls, private conversations and classified documents, including information that helped fuel investigations into alleged Trump campaign collusion with Russia.

The most severe leaks from the State Department and intelligence community that marred the early days of the Trump administration have largely subsided.

The leaks slowed substantially after the Justice Department in June 2017 charged intelligence contractor Reality Leigh Winner with giving classified reports about Russian meddling in the 2016 election to news organizations.

Her trial is scheduled to begin Oct. 15.

No one at the White House has been publicly punished for leaking.

Despite the relatively high rate of turnover among White House officials, divulging information to the media has never been cited as a cause for termination. Karl Rove, former White House chief of staff for President George W. Bush, said Mr. Trump deserved the blame for his leaky staff.

“I’ve never seen a White House leak as much against itself as this one and these are people putting their own personal agendas above those of the country and the president that they serve,” he said on Fox Business’ “Mornings with Maria.” “Frankly, it’s ridiculous, it’s despicable, it’s reprehensible and it does not help anyone.”

He said Mr. Trump set the tone.

“Who’s the biggest leaker in the White House? It’s the president of the United States who calls up media figures all of the time and tells them stuff that he shouldn’t be telling them,” said Mr. Rove, a frequent critic of Mr. Trump.

At the daily press briefing Monday, White House deputy press secretary Raj Shah was peppered with questions about the McCain remark and whether the culprit, Kelly Sadler, would make a public apology or be fired.

He said he understood the news media’s focus on the issue but that it had been handled “internally.”

Mr. Shah did not elaborate about how it was handled, although he said Mrs. Sadler was sill employed at the White House.

He added that Mrs. Sadler apologized to the McCain family before the story was published last week, and he declined to offer further public apologies.

Asked whether Mrs. Sadler was a victim of leaks, as some conservative commentators have asserted, Mr. Shah acknowledged that leaks harmed White House workers’ ability to talk freely.

“If you aren’t able, in internal meetings, to speak your mind or convey thoughts or say anything that you feel without feeling like your colleagues will betray you, that creates a very difficult work environment,” said Mr. Shah.

Mrs. Sadler made the remark last week about Mr. McCain, 81, who is battling brain cancer and has been absent for months from the Senate. It is uncertain whether he will be able to return.

Mrs. Sadler, a former editorial department employee at The Washington Times, made the insensitive comment at a closed-door White House meeting after the senator announced his opposition to CIA director nominee Gina Haspel.

“It doesn’t matter, he’s dying anyway,” Mrs. Sadler said, a participant in the meeting told The Hill.

Mr. McCain has frequently clashed with Mr. Trump. Prior to the brouhaha over the “dying” remark, Mr. McCain reportedly requested Mr. Trump not attend his funeral.

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