Donald Trump’s allies tried their best Sunday to tamp down an eruption of stories accusing the Republican presidential nominee of untoward sexual advances against women, as national polls showed the billionaire businessman still within striking distance of Hillary Clinton after a horrendous week.

His running mate, Indiana Gov. Mike Pence, said he “couldn’t be more proud” to stand shoulder to shoulder with Mr. Trump, despite the rising tally of women who have accused the presidential candidate of groping and other unwanted advances over the past few decades.

“Donald Trump has made it clear that he categorically denies the allegations that have been made against him this week,” Mr. Pence told the CBS program “Face the Nation.”

Former New York Mayor Rudolph W. Giuliani also vouched for the real estate developer.

“I believe my friend Donald Trump when he tells me he didn’t do it,” he told CNN’s “State of the Union” program. “I know Donald. I have been with him for 28 years. I have never seen him do anything like that.”

Mr. Trump has three weeks to make up ground lost in must-win battleground states before voters have their say on Nov. 8.

A Washington Post/ABC News poll released Sunday showed Mr. Trump trailing Mrs. Clinton, the Democratic nominee, by just 4 points, 47 percent to 43 percent — a margin that is relatively unchanged from a similar poll taken ahead of the first presidential debate.

That suggests the fallout from a 2005 “Access Hollywood” tape, in which the mogul speaks about women in lewd terms, and accusations of groping were largely offset by a string of leaked emails that portray Mrs. Clinton’s campaign in an unflattering light.

“I think it’s amazing that Trump is as close as he is right now, considering the one-sidedness of the news media barrage,” former House Speaker Newt Gingrich told ABC’s “This Week.”

Still, nearly seven in 10 of those polled said Mr. Trump probably made unwanted sexual advances toward women, and more than half said his apology for remarks on the hot-mic tape was insincere.

An NBC News/Wall Street Journal poll said Mrs. Clinton was up by 11 points among likely voters, 48 percent to 37 percent, in a poll that incorporates Libertarian Party nominee Gary Johnson and the Green Party’s Jill Stein.

Mrs. Clinton held a 20-point advantage among female voters, and 31 percent of all voters said the presidential debates made them more likely to back the Democrat, compared with 14 percent for Mr. Trump.

The Trump camp’s decision last week to shift resources from Virginia, a state that went for President Obama in 2012, to North Carolina, which Republican nominee Mitt Romney won, underscored his shrinking path to victory.

Against the ropes, Mr. Trump lashed out from the stump by forcefully denying reports that he groped women and even suggested that one of his accusers wasn’t attractive enough to be a target. He insinuated that Mrs. Clinton was on drugs at the last debate and urged his supporters to watch out for funny business at the polls on Election Day.

Mr. Pence, meanwhile, said an avalanche of negative media coverage about his running mate was the real problem.

“We will absolutely accept the result of the election. Look, the American people will speak in an election that will culminate on Nov. 8,” Mr. Pence told NBC’s “Meet the Press.” “But the American people are tired of the obvious bias in the national media. That’s where the sense of a rigged election goes here.”

Mr. Pence said the media should keep delving into emails that show Mrs. Clinton and her staff making unflattering remarks about Catholics and “needy” Hispanics.

At the same time, Mr. Pence said there should be “severe consequences” for overseas hackers trying to meddle in U.S. politics. Investigators say Russians are responsible for the emails published by WikiLeaks, an anti-secrecy organization.

Democratic vice presidential candidate Tim Kaine, a U.S. senator from Virginia, was more forceful about reports that Russia was behind the attacks. “I can’t think of a precedent of a foreign nation trying to destabilize an American election,” he told Fox News.

He said he is confident that Mrs. Clinton’s private email server, which she set up during her tenure at the State Department, was not breached, though he tried to pivot away from the substance of the emails, which he said were illicitly gained.

“I don’t give credence to any of these dumped documents because I don’t even know if they’re accurate,” he told ABC’s “This Week.”

Mrs. Clinton plans to be relatively quiet on the campaign trail in the days leading up to the final presidential debate Wednesday in Las Vegas.

Some analysts insist she is boxed in by the latest reports, given her famous husband’s infidelities, though her surrogates say she should be judged on her own merits.

“It is a double standard. Bill Clinton is not on the ballot,” House Minority Leader Nancy Pelosi, California Democrat, told CNN. “He’s not on the ballot.”

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