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SOUTHAVEN, Miss. (CBS)  — President Trump used his “Make America Great Again” rally in Mississippi on Tuesday night to hit back against the scrutiny surrounding his Supreme Court nominee, Judge Brett Kavanaugh. The confirmation process to appoint Kavanaugh has been delayed over allegations that he engaged in sexual misconduct decades ago.

Mr. Trump mocked the testimony of Dr. Christine Blasey Ford, who is Kavanaugh’s most prominent accusers, after she appeared Thursday before the Senate Judiciary Committee, saying she couldn’t recall key details of the alleged assault.

“How did you get home? I don’t remember,” Mr. Trump said Tuesday, apparently alternating between questioner and an impression of Ford. “How did you get there? I don’t remember. Where is the place? I don’t remember. How many years ago was it? I don’t know.” At that point the crowd erupted in cheers and applause.

Mr. Trump said “a man’s life is shattered,” referencing Kavanaugh, and “these are really evil people.”

Meanwhile, the FBI aims to complete its supplemental background check into the allegations of sexual misconduct made against Kavanaugh by the end of this week, and as Senate Majority Leader Mitch McConnell plans to hold a vote on the nominee in the coming days.

As Republicans fight headwinds ahead of the Nov. 6 midterm election, Mr. Trump rallied his supporters behind Sen. Cindy Hyde-Smith, who was appointed to fill the seat of Republican Thad Cochran, who retired in April. She faces three candidates — Republican Chris McDaniel and Democrats Mike Espy and Tobey Bernard Bartee — in next month’s special election for the remainder of the two-year term.

“She’s always had my back,” Trump said. “She’s always had your back. And a vote for Cindy is a vote for me.”

Republican officials and the White House expect Hyde-Smith’s race to go to a runoff under the state’s jungle election rules that force a showdown between the top two finishers if no candidate wins at least 50 percent of the vote on Election Day. With Republicans defending majorities in the House and Senate next month, officials cast Mr. Trump’s visit as an attempt to get ahead of a potentially perilous situation.

Officials said Mr. Trump is seeking to boost Hyde-Smith as close as possible to the 50 percent threshold and lend momentum for a possible runoff. Depending on how Republicans perform on Nov. 6, the eyes of the nation could fall on a Nov. 27 Mississippi runoff in what could become an expensive and high-profile race to determine control of the Senate.

“Your vote in this election will decide which party controls the United States Senate,” Mr. Trump said.

Mr. Trump’s support for Hyde-Smith is hardly without controversy – even at one of his own rallies. A vocal minority of the crowd on Tuesday backed McDaniel, a conservative state senator, and booed Hyde-Smith when Mr. Trump introduced her. They even launched into occasional chants of “We want Chris.”

Mr. Trump also attacked Espy, a former congressman and the leading Democratic contender, saying a vote for him “is a vote for the Democrat agenda.”

“This is also a referendum about me and the disgusting gridlock they’ll put this country through,” Mr. Trump added of Democrats.

© 2018 CBS Interactive Inc. All Rights Reserved. This material may not be published, broadcast, rewritten, or redistributed. The Associated Press contributed to this report.