Midway through primary season, the party has nominated several candidates who deny the 2020 outcome for posts that will have significant sway over the 2024 presidential election.The potential for far-right Republicans to reshape the election systems of major battleground states is growing much closer to reality.As the halfway point nears of a midterm year that is vastly friendlier to Republicans, the party’s voters have nominated dozens of candidates for offices with power over the administration and certification of elections who have spread falsehoods about the 2020 presidential contest and sowed distrust in American democracy.The only way to restore trust, these candidates say, is by electing them.In Michigan, Pennsylvania and now Nevada, Republican voters have elevated candidates who owe their political rise to their amplification of doubts about Joseph R. Biden Jr.’s victory, and who are now vying in elections for governor, secretary of state and attorney general — offices that will hold significant sway over the administration of the 2024 presidential election in critical swing states.The rise of election deniers is far from over. Primary contests coming later this month in Colorado and in early August in Arizona and Wisconsin will provide more clarity on the depth of Republican voters’ desire to rally behind candidates devoted to the false idea that the 2020 election was stolen from former President Donald J. Trump.With Republicans widely predicted to make gains in November, it is possible that 2023 will bring newly installed far-right officials willing to wield their influence to affect election outcomes and a possible Supreme Court ruling that could give state legislatures unchecked power over federal elections. Even some Republican candidates and officials who for a time defended the 2020 results as legitimate have begun to question whether Mr. Biden’s victory was on the level.“We are in a dangerous place at the moment,” said Ben Berwick, the counsel for Protect Democracy, a nonpartisan group dedicated to resisting authoritarianism. “There is a substantial faction in this country that has come to the point where they have rejected the premise that when we have elections, the losers of the elections acknowledge the right of the winner to govern.”On Tuesday, Nevada Republicans chose as their nominee for secretary of state Jim Marchant, an organizer of a Trump-inspired coalition of far-right candidates united by their insistence that the 2020 election was rigged. Mr. Marchant, a former state legislator from Las Vegas, told voters during a February debate that “your vote hasn’t counted for decades.”Doug Mastriano, the Republican nominee for governor of Pennsylvania, was a leading figure in the effort to subvert the state’s 2020 results on behalf of former President Donald J. Trump.Michael M. Santiago/Getty ImagesOn the November ballot, Mr. Marchant joins Doug Mastriano, the Republican nominee for governor of Pennsylvania, who won his primary last month after promoting efforts to decertify the 2020 results, and Attorney General Ken Paxton of Texas, who in December 2020 sued to overturn Mr. Biden’s victory, as well as like-minded figures in other states including New Mexico.The number of election deniers who have won Republican nominations is quickly rising in congressional and state legislative races across the country. At least 72 members of Congress who voted to overturn the 2020 election have advanced to the general election, according to a New York Times analysis.Understand the June 14 Primary ElectionsTakeaways: Republicans who embraced former President Donald J. Trump’s election lies did well in Nevada, while his allies had a mixed night in South Carolina. Here’s what else we learned.Winners and Losers: Here is a rundown of some of the most notable wins and losses.Election Deniers Prevail: Republicans who deny the 2020 election’s result are edging closer to wielding power over the next one.Nevada Races: Trump-inspired candidates captured key wins in the swing state, setting the stage for a number of tossup contests against embattled Democrats.Texas Special Election: Mayra Flores, a Republican, flipped a House seat in the Democratic stronghold of South Texas. Her win may only be temporary, however.And in Georgia, Pennsylvania, North Carolina and Texas — four competitive states that have already held primaries — 157 state legislators who took concrete steps to overturn or undermine the 2020 election will be on the ballot in November.These primary results come as the House select committee’s hearings into the Jan. 6 attack on the Capitol have revealed how nearly every senior figure in the Trump orbit except the president himself believed Mr. Biden had won the election despite Mr. Trump’s claims.Still, the former president’s lies, adopted and advanced by his followers, continue to threaten to upset the country’s democratic order nearly 18 months later.For many election-denying candidates, victory is far from assured. Some of the most prominent ones, like Mr. Mastriano, face tough general-election campaigns, and their success may depend on factors like their personal fund-raising networks, the health of the economy and policy debates that have nothing to do with election administration.And voters have at times been hesitant to embrace candidates whose central plank is elections. Gov. Brian Kemp of Georgia swept aside former Senator David Perdue’s 2020-centric challenge, and Brad Raffensperger, the state’s secretary of state, also handily beat a Trump-backed rival.Still, in primary after primary, election deniers have ascended, signaling that Mr. Trump’s falsehoods about the 2020 election have become deeply embedded in the Republican base.Voters waiting to enter a temporary polling location in Las Vegas on Tuesday.Bridget Bennett for The New York TimesHanging over several marquee 2022 races is the prospect of 2024, when a Democratic presidential nominee — Mr. Biden, if he runs again as promised — might have to confront the open question of whether victories in certain states would be certified.In several battleground states, Republicans who have said they would not have certified Mr. Biden’s 2020 victory are running for governor or secretary of state, positions that oversee elections and the appointment of Electoral College delegates.In particular, races for secretary of state — once little-noticed contests to choose the top election official in most states — have become extraordinarily high-profile and politicized.“I don’t know of a single proven competent election official that says, ‘Gosh, I can’t wait to be on the front page,’” said Pam Anderson, a Republican running for secretary of state in Colorado. “Because usually that’s a really bad thing.”Ms. Anderson is running in a Republican primary against Tina Peters, the Mesa County clerk, who is under indictment related to allegations that she tampered with elections equipment after the 2020 election. Ms. Peters has become something of a hero to the far-right base, walking the red carpet at a documentary screening at Mar-a-Lago and speaking at Republican events across the country.Tina Peters speaking at a rally hosted by Mr. Trump in Casper, Wyo., in May.Natalie Behring for The New York TimesKristina Karamo addressing the crowd at a Trump rally in Washington, Mich., in April.Brittany Greeson for The New York TimesPromoting 2020 falsehoods has also bolstered the prospects of candidates for secretary of state who have no experience managing elections, like Kristina Karamo, who is the likely Republican nominee in Michigan after winning the most delegates at the state party’s convention.Understand the 2022 Midterm ElectionsCard 1 of 6Why are these midterm races so important? More