The party is targeting states with Democratic governors but overwhelming Republican legislative control, effectively battling to win back veto power.Democrats are planning to spend millions of dollars next year on just a few state legislative elections in Kansas, North Carolina, Kentucky and Wisconsin — states where they have little to no chance of winning control of a chamber.Yet what might appear to be an aimless move is decidedly strategic: Democrats are pushing to break up Republican supermajorities in states with Democratic governors, effectively battling to win back the veto pen district by district. Such supermajorities result when a single political party has enough votes in both chambers of a legislature to override a governor’s veto, often, though not always, by controlling two-thirds of the chamber.The extraordinary political dissonance of having a governor of one party and a supermajority of an opposing party in the legislature is one of the starkest effects of gerrymandering, revealing how parties cling to evaporating power.As gerrymanders built by both parties for decades have tipped the scales to favor the party of the map-drawers, legislative chambers have proved resistant to shifting political winds at the state level. At times, those gerrymanders have locked in minority rule in legislatures while statewide offices, like the governor’s, adhere to the desires of a simple majority of voters.Though both parties employed aggressive gerrymanders during the last round of redistricting in 2021, Republicans entered the cycle with a distinct advantage: In 2010, G.O.P.-controlled state legislatures across the country drew aggressive gerrymanders in state governments. Democrats were caught off guard.“The bottom fell out,” said Heather Williams, the interim president of the Democratic Legislative Campaign Committee. “And we’ve been building back since then.”As a result, Republicans now control resilient supermajorities in Kansas, North Carolina and Kentucky, even as Democrats hold the executive branch. And in Wisconsin, Republicans control a supermajority of the State Senate, which can act unilaterally on issues like impeachment, and are just two seats shy of a supermajority in the State Assembly, though last year Gov. Tony Evers, a Democrat, won re-election.The Democratic Legislative Campaign Committee has committed “more than seven figures” of its initial $60 million budget for 2024 to breaking up these four supermajorities, with the caveat that redistricting efforts in North Carolina and Wisconsin could shift resources. “Republicans in these legislatures are not moderate,” Ms. Williams said. “They are governing very extremely, and we need a stopgap, and it is critical that governors have veto power where their legislature and their legislative maps are so gerrymandered.”The only example where the parties are flipped is in Vermont, where a Democratic supermajority in the legislature overrode multiple vetoes by Gov. Phil Scott, a Republican, this year. And in Nevada, Democrats control a supermajority of the State Assembly and are just one vote shy of a supermajority in the State Senate, while Gov. Joe Lombardo, a Republican, was elected in 2022.A spokesman for the Republican State Leadership Committee did not respond to questions about similar strategies for Republicans.Though Democrats have occasionally ventured into conservative-leaning legislative districts, such an extensive foray into fairly hostile territory will be a new challenge, particularly in deeply red states like Kansas where Democratic voters are often ignored during better-funded national campaigns for president. Recruiting candidates to serve in the minority, rather than to play a role in flipping a chamber — a more energizing prospect — can also pose a challenge.But while state legislative elections are often defined by issues as hyperlocal as a traffic intersection or funding for an after-school program, Democrats are also hoping that one critical national issue will help them: abortion.Despite President Biden’s persistent unpopularity, Democrats last week took back the Virginia General Assembly and won the governor’s race in deep-red Kentucky, as well as a majority of this year’s special elections, largely because abortion access was a motivating issue.On the heels of the Supreme Court’s decision to overturn Roe v. Wade last year, Kansas voters rejected an amendment that would have effectively eliminated abortion in the state. But in the Legislature, dominated by Republicans, “we had 21 different bills come up in committee trying to restrict abortion access,” said Jeanna Repass, the chair of the Kansas Democratic Party. “So what that has taught us is that if we can get the messaging out to people, we can get them interested in the fact that they’re not being represented by their legislators.”“When I’m out, I hit them hard with abortion, our public schools and Medicaid, and in that order,” Ms. Repass added.As Democrats invest in trying to climb out of superminority positions, they will face some deep-pocketed state Republicans. Robert Reives, the Democratic minority leader in the North Carolina General Assembly, pointed to two races in 2022 that featured Republican candidates spending roughly $800,000 each to defeat Democratic incumbents.“They had the benefit of having two billionaires that kind of financed a lot of the top line of the campaign and then just kind of went from there,” Mr. Reives said. “Unfortunately, we don’t have billionaires on our side to do that.”Mr. Reives was confident that even with newly drawn maps favoring Republicans, the Democrats would have a chance of breaking the supermajority in the state in 2024, focusing on urban areas like Wake County, home to Raleigh. And he said that while abortion would inevitably be a factor in coming elections, the hyperlocal issue of authorizing casinos in the state is likely to help Democrats claw back a few seats.“They were literally going against every constituency,” Mr. Reives said, referring to broad opposition to casino expansion. Even some Republicans objected to it.One path for Democrats to win back their veto pens can be found in eastern Wisconsin.In 2022, Democrats stared down gerrymandered maps that raised the possibility of a Republican supermajority even as Mr. Evers, the Democratic governor, cruised to a re-election victory.As returns trickled into the party headquarters in Madison, party officials breathed a sigh of relief when Steve Doyle, a 10-year incumbent from La Crosse, defeated his Republican challenger by 756 votes. His race was won not on the airwaves or even necessarily just on the issues, but on the pavement, as Mr. Doyle undertook an extensive door-knocking campaign to meet all of his voters, according to Greta Neubauer, the Democratic minority leader in the Wisconsin Assembly.“This is a Trump-won district that Democrats at the top of the ticket struggle to win,” Ms. Neubauer said. “But he spends a lot of time on his acquisition of voters, and constantly fending off attempts to take him out.” More