The Trump administration went on the offensive Sunday to beat back purported advances in the Democrats’ impeachment inquiry and bipartisan outrage over the president’s military pullout in Syria, events that last week knocked the White House on its heels.

Even with both parties at war over the impeachment inquiry, President Trump’s decision to withdraw troops from northern Syria, though they will reportedly stay in the region to counter the Islamic State, united lawmakers across Capitol Hill in condemnation.

Despite the backlash from Capitol Hill, Secretary of State Mike Pompeo was optimistic that the administration’s strategy would pay off.

He pushed back Sunday against reports that the cease-fire he and Vice President Mike Pence negotiated last week was a victory for Turkey and said efforts to “crush” ISIS would continue to be a top priority.

“I got a report within the last half hour from my senior leaders who indicate there is relatively little fighting, little sporadic small arms fire and a mortar or two,” Mr. Pompeo said Sunday on ABC’s “This Week.”

The cease-fire was met with skepticism by top lawmakers, including House Speaker Nancy Pelosi, California Democrat, and Sen. Lindsey Graham, South Carolina Republican, who called the commitment flimsy and said it left America’s Kurdish allies in the lurch. The House passed an overwhelmingly bipartisan rebuke of the decision Wednesday, though Senate Majority Leader Mitch McConnell, Kentucky Republican, said he wanted a stronger version from his chamber.

Mr. Graham, one of the president’s top critics on the issue, had a different attitude Sunday.

Speaking on Fox News, he said he was still concerned about the U.S. commitment to the endangered Kurds but was optimistic about Mr. Trump’s plan to move troops to secure the oil fields.

He said it was an opportunity for a “historic” way forward for all parties involved by having Syrian Democratic Forces join a “small force” of Americans to keep the oil resources out of the hands of Syria’s regime and its Iranian allies.

“The bottom line here is, you have got to play the ball where it lies. And if President Trump fulfills the objectives he laid out to me, then I think we can end Syria successfully,” he said.

House Minority Leader Kevin McCarthy, a California Republican who voted to condemn Mr. Trump’s Syria decision, said on Fox News that he didn’t think Congress should take any more action, including moving forward on sanctions against Turkey, until the administration had a chance to brief lawmakers on their progress abroad.

While Syria was straining relations, Republicans were in a tough spot with reports that closed-door impeachment inquiry sessions did not bode well for the president.

Over the past three weeks, several State Department officials have testified as part of Democrats’ impeachment inquiry centered on accusations that Mr. Trump pressured Ukrainian leaders to open investigations for his personal political gain.

Reports on those confidential meetings showed that several officials were concerned about the Ukraine policy role of Mr. Trump’s personal attorney Rudolph W. Giuliani, who was pursuing an investigation into the Biden family. They were also concerned about the abrupt ouster of Marie Yovanovitch as ambassador to Ukraine, who reportedly resisted the administration’s efforts to have Ukraine open investigations.

She was later disparaged by Mr. Trump and Mr. Giuliani and became a target in a conspiracy theory packet delivered to Mr. Pompeo.

The secretary said he did pass along that packet but never reviewed the contents.

Acting White House Chief of Staff Mick Mulvaney made matters worse for the president when he publicly linked the nearly $400 million in delayed military aid for Ukraine in part to the administration’s desire for an investigation into the 2016 election and whether Democratic National Committee servers were located in the Eastern European country.

He later walked back his remarks and said the news media misconstrued them to further their “witch hunt.”

During his interview, Mr. Pompeo wouldn’t comment on Mr. Mulvaney’s remarks but said he never saw anything concerning about Ukraine.

“The conversation was always around what were the strategic implications,” he said about the delayed aid.

He did, however, defend Mr. Giuliani’s role in the Ukraine situation.

“Private citizens are often part of executing American foreign policy,” he said. “This is completely appropriate.”

The former New York City mayor defied a congressional subpoena Tuesday to turn over documents related to the Ukraine investigation.

Mr. Giuliani has become one of the central figures in the impeachment inquiry as he works to push Ukraine to open an investigation corruption involving Vice President Joseph R. Biden, one of the 2020 Democratic presidential front-runners, and his son Hunter Biden.

Rep. James A. Himes, Connecticut Democrat, and Rep. Will Hurd, Texas Republican, who sit on the House Permanent Select Committee on Intelligence, said Mr. Giuliani should speak directly to Congress about his role in the situation.

“While I guess we could sort of piece together exactly what happened based on what the administration has disclosed, I really think it’s important to talk to Rudy Giuliani,” Mr. Himes said on CBS’s “Face the Nation.”

Mr. Mulvaney continued to walk back the comments he made at Thursday’s press conference after Fox News host Chris Wallace pointed out two examples of him linking aid to an investigation into 2016 election interference in Ukraine.

“I never said there was a ‘quid pro quo’ ‘cause there isn’t,” he said, arguing that the proof was that the aid was eventually delivered.

On Capitol Hill, Republicans have continued to push back on the inquiry as an illegitimate, secretive and unfair process.

They are set to force a vote to try to censure House intelligence Committee Chairman Adam B. Schiff, California Democrat, and have sent letters to chairmen spearheading the impeachment investigation demanding that their rights to the committee materials be honored, citing the precedents of the impeachment cases against Democrat Bill Clinton and Republican Richard Nixon.

“You want to trust democracy. And from a very important standpoint here, we need due process in this. A Republican speaker made sure this process happened with President Clinton. A Democrat speaker made sure this process happened with a Republican president,” Mr. McCarthy said Sunday.

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