It was only on Sunday evening that LIV Golf, the men’s league awash in billions of dollars from Saudi Arabia’s sovereign wealth fund, met its greatest athletic triumph to date when one of its headliners, Brooks Koepka, emphatically won the P.G.A. Championship.By Thursday morning, though, LIV’s road show had been reinfused with the political bent that has trailed the second-year circuit as it has convulsed professional golf: the loquacious, limelight-seizing presence of former President Donald J. Trump, who is hosting one of the league’s tournaments this weekend at a course northwest of Washington.Whether LIV can outrun Trump’s shadow, and whether it even wants to, could do much to shape how the league is perceived in the years ahead, particularly in the United States, where it has struggled to gain a meaningful foothold against the PGA Tour.But for now, besides major tournament winners like Koepka and Phil Mickelson who have joined the circuit, there is probably no figure beyond golf more publicly linked to LIV than Trump, who has repeatedly and enthusiastically cheered Saudi Arabia’s thunderous, flashy entrance into sports. At its events, he often seems like an eager M.C. whose role is at once decidedly conspicuous and deeply mysterious — neither the Trump Organization nor LIV have disclosed how much money the former president’s company is making for the events — as the league looks to make inroads in a hidebound sport.“They want to use my properties because they’re the best properties,” Trump said on Thursday, when he spent five hours appearing in a pro-am event with the LIV players Graeme McDowell and Patrick Reed (and staging what amounted to a rolling news conference about politics and an infomercial about his property over 18 holes along the Potomac River).The Trump portfolio does indeed feature some exceptional courses, including the Washington-area location, which once held a Senior P.G.A. Championship, and LIV executives have said in the past that they were drawn to them because many top-caliber properties in the United States were not willing to host a circuit intended to rival the PGA Tour. But Trump’s persistent, growing place in LIV’s orbit also invites sustained skepticism of the motives and intentions of the league, which some critics see as a glossy way for Saudi Arabia to rehabilitate its image.The former president is unbothered by the league’s patron, Saudi Arabia’s sovereign wealth fund, and the kingdom’s budding place in professional golf, despite its record of human rights abuses. He is still casting aside objections from family members of Sept. 11 victims, some of whom believe Saudi Arabia played a role in the 2001 attacks, because, as he said Thursday, LIV tournaments are “great economic development.” He is openly admiring the millions and millions of dollars that the Saudis are raining down onto players and, of course, properties like his, even though he asserted Thursday that hosting tournaments amounts to “peanuts for me.” This year, LIV will travel to three of his properties, up from two in its inaugural season.He has remained steadfast in his loyalty even though a special counsel from the Justice Department, Jack Smith, has subpoenaed the Trump Organization for records related to LIV.In an interview as he walked between holes on Thursday, Trump described Smith’s aggressive approach as “retribution” because the Biden administration wants “to do something to take the spotlight off what’s taken place.” He said he did not know why his ties to LIV had drawn the special counsel’s scrutiny.Trump’s affection for LIV can be traced, at least in part, to years of friction with golf’s establishment.In 2016, the PGA Tour ended a longstanding relationship with Trump’s course in Doral, Fla., near Miami, because of what its then-commissioner described as “fundamentally a sponsorship issue.” And in 2021, after Trump supporters stormed the Capitol, the P.G.A. of America — which is separate from the PGA Tour — abandoned its plan to host its flagship men’s championship at a Trump property in New Jersey in 2022.Trump has not fared much better abroad. The R&A, which organizes the British Open, has signaled it does not intend to take the tournament back to Trump-controlled Turnberry, where LIV’s commissioner, Greg Norman, won one of his two Opens.LIV has embraced Trump, though, and in return gotten a former president’s imprimatur, along with bursts of news coverage for events that might have gone unnoticed otherwise. He brings prestige and power, diluted as both might be by the divisiveness in which he revels.“They have unlimited money and they love it,” he said Thursday, “and it’s been great publicity for Saudi Arabia.”But for every day Trump appears at a LIV event, it is a day that LIV might as well write off as one in which it will not escape the pointed questions that it has spent a year trying to move past, or at least saying it wants to move past.It has been hard enough for the league, even on a day when Trump is not playing a round, not to have its players confronting questions about the morality of accepting millions in Saudi money.“We’re contracted to play golf,” Bryson DeChambeau, the 2020 U.S. Open winner who finished in a tie for fourth at the P.G.A. Championship last weekend, said on Wednesday. “I think the most important part is to provide great entertainment wherever possible on whatever platform that is, whatever platform that provides it. When you can talk about ethics, that’s people’s perception. I completely disagree with it, but everybody has the right to their own opinion, and I’d say, was it worth it? Absolutely.”But DeChambeau hardly has the same megaphone or presence as a former occupant of the Oval Office. When Trump appears at a LIV event, even winners of the Masters Tournament or the U.S. Open are relegated to supporting actors.LIV executives have generally brushed aside questions about whether the former president is good for business, or merely essential for it, given their troubles landing quality venues. They seem convinced that, at some point, sports will overtake politics, which might be wishful thinking since Trump suggested Thursday that nothing — not even a return to the White House — would easily dissuade him from doing business with the league.But LIV’s strategy still involves a gamble that the presence of one of the nation’s most polarizing figures will not scare off even more of the sponsorship contracts and television rights that are already proving hard to come by for the operation. And Trump can just as easily alienate prospective fans as he can entice them.Trump himself insists that LIV craves him at its events and that he is not a distraction from the league’s proclaimed goal of growing the sport and giving it doses of needed energy.“They wanted me to be here, and I said sure,” said Trump, who said that LIV’s contracts with his properties did not require his appearances in events like the pro-am.Perhaps all of that is true. But as long as it is, LIV will linger in the political thicket, no matter how well Koepka plays on the game’s biggest stages.