A growing number of contests to elections in Texas’ Harris County are a broad attempt to cast doubt on an Election Day that officials concede had problems.HOUSTON — Jon Rosenthal has seen some close races, but his re-election to the Texas State House in November, in a Houston district redrawn to be a virtual lock for Democrats, was not one of them. Mr. Rosenthal won by 15 points.So it came as a surprise when his Republican challenger in the race contested the results, petitioning the State Legislature to order a new election.Another surprise came late Thursday when the Republican candidate for the top executive position in Harris County, which includes Houston, announced that she, too, would contest her much narrower loss, by about 18,000 votes, to the progressive Democrat who is the county’s incumbent chief executive, Lina Hidalgo. By Friday, more than a dozen losing Republican candidates had filed suits to contest the results of their races.Election Day in Harris County, Texas’ largest county, saw a range of problems at polling places, including some that opened late and others that ran out of paper for printing voted ballots. A court ordered the polls to stay open an extra hour to compensate; then the Texas Supreme Court stepped in and halted the extra voting.Republicans, who have been watching closely for election issues in races around the country, seized on the difficulties in Harris County, which is becoming a Democratic stronghold. Candidates called into question the reliability of the results in a bitter and expensive campaign that failed to dislodge Ms. Hidalgo and a slate of Democratic judges.“It is inexcusable that after two months, the public is no further along in knowing if, and to what extent, votes were suppressed,” said Alexandra del Moral Mealer in explaining her decision to contest her loss to Ms. Hidalgo, adding that her challenge was “fundamentally about protecting the right to vote in free and fair elections.”Candidates called into question the reliability of the results in a bitter campaign that ended with Republicans failing to oust Lina Hidalgo and a slate of Democratic judges. Annie Mulligan for The New York TimesElection contests are not uncommon in Texas, often involving down-ballot races in small counties where the margins are often notably slim. But the challenges in Harris County appeared to be uniquely broad in their attempt to cast doubt on much of the voting process in an election that involved 1.1 million votes. They followed calls from state leaders, including Gov. Greg Abbott, for an investigation into the county’s handling of the election. The local district attorney opened an inquiry in November.The election contests in Harris County have at times resembled the one mounted in Arizona by the Republican candidate for governor, Kari Lake, who has sought to overturn her loss by claiming that election officials in one major county deliberately disenfranchised her voters. A judge dismissed her claims last month for lack of evidence.But the latest contests in Texas have little precedent, said the Harris County attorney, Christian Menefee, a Democrat. “To my knowledge, this is the first election contest filed in Harris County that is wholly focused on these kinds of process failures,” Mr. Menefee said in an interview.The sprawling Texas county has shifted more decisively in the direction of Democrats in the last few election cycles, following in the direction of other major Texas population centers.For a variety of reasons, it has struggled to conduct elections smoothly, drawing repeated scrutiny from Republican lawmakers in the State Capitol. The county’s size has been a challenge, covering an area nearly the size of Delaware with 2.5 million registered voters and more than 700 polling places. It has struggled with newly mandated voting systems and has not had steady leadership at its elections office, with three different administrators since 2020.An audit of the 2020 election, conducted by the secretary of state, highlighted a range of issues, including instances where Harris County did not handle its electronic records properly, though there was no evidence of widespread fraud.Several steps that the county took during the coronavirus pandemic to make it easier to vote in Houston — such as limited 24-hour voting and drive-through polling places — also drew criticism from Republicans, who argued that the changes had made the election less secure. The Republican-dominated State Legislature, in its last session, took steps to curtail many of the measures.Voters waited in line at Damascus Missionary Baptist Church on Election Day in Houston in November.Annie Mulligan for The New York TimesOn Election Day in November, the county experienced problems at a number of polling places, including several that were significantly delayed in opening and others that reported running out of paper ballots.A judge ordered polling places in the county to remain open for an extra hour after the Texas Organizing Project, a nonprofit, filed suit over the issues, claiming that voters were being prevented from casting ballots. The Texas Supreme Court stepped in and stayed the ruling in response to a challenge from the Republican attorney general, Ken Paxton. The court eventually allowed about 2,000 provisional ballots that were cast during the extra voting time to be added to the official count.The county elections administrator, Clifford Tatum, has defended the election process and said the issues that came up reflected small problems in an otherwise well-run election. “Overall, Election Day was a success,” a postelection report from Mr. Tatum’s office concluded.But the report, released last week, also found that the county’s voting system was in “an immediate need of upgrades or replacements” to correct software issues, simplify voting day setup and create a system for the elections administrator to know in the moment whether problems reported at polling places had been resolved.The Harris County Republican Party has focused on a broad range of issues that arose on Election Day, including not only sites that ran out of paper ballots but also others where poll workers incorrectly fed paper ballots into the voting machines.In its report, the election administrator’s office said that officials at 68 voting centers reported running out of the initial allotment of paper on Election Day, and that only 61 of them said they had received deliveries of more paper.But it remained unclear how many voters were turned away because of the paper shortages, in part because, according to the report, some of the election judges “declined to speak after reportedly being advised not to do so by the Harris County Republican Party.”A spokeswoman for the county Republican Party, Genevieve Carter, denied any such instructions. “We encouraged them to provide their firsthand account of any issues that occurred,” she said. “Our goal is to get to the bottom of what went wrong during this election.”The party’s lawyers and leaders have not claimed that they can prove their candidates should have won. Instead, they have argued that the scope of the problems on Election Day were so great — including, they claimed, allowing some voters to cast ballots who were no longer eligible to do so in the county — that the true results in the election cannot be known; they are demanding that new elections be held. (More than two-thirds of the ballots were cast either during early voting or by mail, not on Election Day.)“We have a systematic cancer that has invaded our election process,” said the chair of the county Republican Party, Cindy Siegel.Democrats have not raised public challenges, but have privately complained that the repeated issues in the election process in Houston were not being adequately addressed, giving Republicans fuel for their efforts to pass new restrictive laws and, now, election contests.Jon Rosenthal said he believed the challenge to his election was frivolous, and that allowing it to go forward in the State House could cause future headaches for lawmakers. Kevin Dietsch/Getty ImagesOnly the candidates themselves can initiate the contests, and so far at least 14 have done so, including Ms. Mealer, a first-time candidate who received millions in campaign contributions from top Houston-area donors; Mr. Rosenthal’s challenger, Michael May; a candidate for county district clerk; and nine Republican judicial candidates.One of the earliest challenges came from a judicial candidate, Erin Lunceford, who lost by 2,743 votes, and filed suit late last year. Ms. Lunceford’s suit includes 19 separate claims of issues with the way votes were handled or counted during the November election and asks the court to void the judicial election and “declare that the true outcome of the election cannot be ascertained.” Ms. Lunceford is represented by Andy Taylor, an election lawyer for the county Republican Party.Ryan MacLeod, a lawyer for the Democrat who won the race, Tamika Craft, described the suit in court papers as a “stunt to make headlines” after an election was lost, and said that “no allegations are supported by facts” and that no evidence had been provided.In the latest challenge on Thursday to the outcome of the race for Harris County judge — effectively the county’s chief executive — Ms. Mealer’s lawyers focused primarily on the paper ballot issues, arguing that they had been concentrated in high-turnout Republican areas and that county officials had “suppressed the voting rights” of residents in those places.Ms. Hidalgo’s office referred questions to the county attorney, Mr. Menefee, who described the challenges as “frivolous attempts to overturn the votes of more than a million residents.”Unlike the other challenges, Mr. May’s contest to his loss against Mr. Rosenthal does not go before a judge, because it involved a State House race. Instead, under Texas law, it will be considered by state legislators, who reconvene this month. The House could decide that the challenge is frivolous and reject it quickly, or choose to investigate the allegations by gathering testimony and evidence before deciding whether the result should be voided and a new election held.Mr. May, in his petition, cited the paper ballot issues and argued that eligible voters were turned away and unable to cast ballots. He has not provided evidence and did not respond to a request for comment.Mr. Rosenthal said he believed the challenge was frivolous and that allowing it to go forward could cause future headaches for lawmakers.“If there is life given to this, and there is no consequence for bringing something this frivolous, you’re setting up for election challenges across the state,” he said. “You could have dozens of challenges per cycle.” More