Among the races to watch are governors’ contests in Kentucky, Louisiana and Mississippi and mayoral elections in Chicago and Philadelphia.It might be tempting to focus on the 2024 presidential election now that the midterms are in the rearview mirror, but don’t sleep on 2023: key races for governor, mayor and other offices will be decided.Their outcomes will be closely watched for signs of whether Democrats or Republicans have momentum going into next year’s presidential election and congressional races — and for what they signal about the influence of former President Donald J. Trump.Virginia and New Jersey have noteworthy state house elections, and in Wisconsin, a state Supreme Court race will determine the balance of power in a body whose conservative majority routinely sides with Republicans. Here’s what to watch:Kentucky governorOf the three governors’ races this year, only Kentucky features an incumbent Democrat seeking re-election in a state that Mr. Trump won in 2020. The race also appears packed with the most intrigue.Gov. Andy Beshear won by less than 6,000 votes in 2019, ousting Matt Bevin, the Trump-backed Republican incumbent in the cherry-red state that is home to Senator Mitch McConnell, the Senate G.O.P. leader.A growing field of Republicans has ambitions of settling the score in 2023, including Daniel Cameron, who in 2019 became the first Black person to be elected as Kentucky’s attorney general, an office previously held by Mr. Beshear. Mr. Cameron, who is seen as a possible successor to Mr. McConnell, drew attention in 2020 when he announced that a grand jury did not indict two Louisville officers who shot Breonna Taylor. Last June, Mr. Trump endorsed Mr. Cameron for governor, but there will be competition for the G.O.P. nomination.Attorney General Daniel Cameron, signing the papers for his candidacy last week, is among Republicans seeking to challenge Gov. Andy Beshear this year.Timothy D. Easley/Associated PressKelly Craft, a former ambassador to the United Nations under Mr. Trump, is also running. So are Mike Harmon, the state auditor of public accounts, and Ryan Quarles, the state’s agricultural commissioner, and several other Republicans. The primary will be on May 16.Louisiana governorGov. John Bel Edwards, a Democrat who narrowly won a second term in 2019, is not eligible to run again because of term limits. The open-seat race has tantalized some prominent Republicans, including Jeff Landry, the state’s attorney general, who has declared his candidacy.Two other Republicans weighing entering the race are John Schroder, the state treasurer who has told supporters he will run, and Representative Garret Graves.Shawn Wilson, the state’s transportation secretary under Mr. Edwards, is one of the few Democrats who have indicated interest in running in deep-red Louisiana.Electing a New Speaker of the HouseRepresentative Kevin McCarthy won the speakership after a revolt within the Republican Party set off a long stretch of unsuccessful votes.Inside the Speaker Fight: Mr. McCarthy’s speaker bid turned into a rolling disaster. “The Daily” has the inside story of how it went so wrong and what he was forced to give up.A Tenuous Grip: By making concessions to far-right representatives, Mr. McCarthy has effectively given them carte blanche to disrupt the workings of the House — and to hold him hostage to their demands.Looming Consequences: Congressional gridlock brought on by far-right Republicans now seems more likely to lead to government shutdowns or, worse, a default on debt obligations.Roots of the Chaos: How did Mr. McCarthy’s bid become a four-day debacle? The story begins with the zero-sum politics of Newt Gingrich.Mississippi governorGov. Tate Reeves, a Republican, is running for a second term. But the advantage of incumbency and a substantial campaign fund may not be enough to stop a primary challenge, especially with his job approval numbers among the lowest of the nation’s governors.Philip Gunn, Mississippi’s House speaker, has been coy about possible plans to enter the race after announcing in November that he would not seek re-election to the Legislature. Among the other Republicans whose names have been bandied about is Michael Watson, the secretary of state. But Mr. Reeves is the only Republican to have filed so far; the deadline is Feb. 1.A Democrat hasn’t been elected governor of Mississippi in two decades, since a contest was decided by the Legislature because the winning candidate did not receive a majority of votes. Not surprisingly, few Democrats have stepped forward to run. One name to watch is Brandon Presley, a public service commissioner. Mr. Presley is a relative of Elvis Presley, who was from Tupelo, Miss., according to Mississippi Today, a nonprofit news website.U.S. House (Virginia’s Fourth District)The death in late November of Representative A. Donald McEachin, a Democrat from Virginia, prompted Gov. Glenn Youngkin, a Republican, to schedule a special election for Feb. 21.In December, Democrats resoundingly nominated Jennifer McClellan, a state senator, to represent the party in the contest for Virginia’s Fourth District, which includes Richmond and leans heavily Democratic. She could become the first Black woman elected to Congress in Virginia, where she would complete the two-year term that Mr. McEachin won by 30 percentage points just weeks before his death.Republicans tapped Leon Benjamin, a Navy veteran and pastor who lost to Mr. McEachin in November and in 2020.Chicago mayorMayor Lori Lightfoot of Chicago, a Democrat who in 2019 became the first Black woman and first openly gay person to lead the nation’s third-most populous city, faces a gantlet of challengers in her quest for re-election.That test will arrive somewhat early in the year, with the mayoral election set for Feb. 28. If no candidate finishes with a majority of the votes, a runoff will be held on April 4.Mayor Lori Lightfoot of Chicago faces several challengers in her re-election bid.Jim Vondruska/ReutersThe crowded field includes Representative Jesús G. García, a Democrat who is known as Chuy and who was overwhelmingly re-elected to a third term in his Cook County district in November and previously ran unsuccessfully for mayor. In the current race, Ms. Lightfoot has attacked Mr. García over receiving money for his House campaign from Sam Bankman-Fried, the criminally charged founder of the collapsed cryptocurrency exchange FTX.Ms. Lightfoot’s other opponents include Kam Buckner, a state legislator; Brandon Johnson, a Cook County commissioner; Sophia King and Roderick T. Sawyer, who both serve on the City Council; Paul Vallas, a former chief executive of Chicago public schools; and Ja’Mal Green, a prominent activist in the city.Philadelphia mayorAn open-seat race for mayor in Pennsylvania’s foremost Democratic bastion has attracted an expansive field of candidates. The office is held by Jim Kenney, a Democrat who is not eligible to run again because of term limits.Five members of the City Council have resigned to enter the race, which city rules require. They are Allan Domb, Derek Green, Helen Gym, Cherelle Parker and Maria Quiñones Sánchez.The field also includes Rebecca Rhynhart, the city’s controller, who has likewise resigned in order to run; Amen Brown, a state legislator; Jeff Brown, a supermarket chain founder; and James DeLeon; a retired judge.Wisconsin Supreme CourtConservatives are clinging to a one-seat majority on Wisconsin’s Supreme Court, but a retirement within the court’s conservative ranks could shift the balance of power this year. The court’s justices have increasingly been called on to settle landmark lawsuits involving elections, gerrymandering, abortion and other contentious issues.Two conservative and two liberal candidates have entered what is technically a nonpartisan election to succeed Judge Patience D. Roggensack on the seven-member court.Daniel Kelly, a conservative former justice on the state Supreme Court who lost his seat in the 2020 election, is seeking a comeback. Running against him in the conservative lane is Jennifer Dorow, a circuit court judge in Waukesha County who drew widespread attention when she presided over the trial of Darrell E. Brooks, the man convicted in the killing of six people he struck with his car during a Christmas parade in Waukesha, Wis., in 2021.Janet Protasiewicz and Everett Mitchell, judges from Milwaukee County and Dane County, which includes Madison, the capital, are seeking to give liberals a majority on the court.The two candidates who receive the most votes in the nonpartisan primary on Feb. 21 — regardless of their leanings — will face each other in the general election on April 4.Legislature (Virginia and New Jersey)Virginia is emerging as a potential tempest in 2023, with its divided legislature up for re-election and elected officials squarely focused on the issue of abortion — not to mention a Republican governor who is flirting with a run for president.Gov. Glenn Youngkin wants to ban most abortions after 15 weeks of pregnancy, emboldened by the Supreme Court’s repeal last summer of Roe v. Wade, the 50-year-old constitutional right to an abortion.His proposal is expected to resonate with Republican lawmakers, who narrowly control the House of Delegates. But it is likely to run into fierce opposition in the Senate, where Democrats are clinging to a slender majority. All seats in both chambers are up for election.Another Mid-Atlantic state to watch is New Jersey, where Republicans made inroads in 2021 despite being in the minority and are seeking to build on those gains.