FRANKFORT, Ky. — Gov. Andy Beshear has been toting oversize checks around his state in recent weeks, handing them out to city and county officials for desperately needed water improvements.

The tiny city of Mortons Gap got $109,000 to bring running water to six families who do not have it. The people of Martin County, whose water has been too contaminated to drink since a coal slurry spill two decades ago, got $411,000. The checks bear Mr. Beshear’s signature, but the money comes from the federal government, part of a huge infusion of coronavirus relief aid that is helping to fuel record budget surpluses in Kentucky and many other states.

Therein lies a Washington controversy. The funds, which Congress approved at a moment when the pandemic was still raging, are allowed to be used for far broader purposes than combating the virus, including water projects like those in Kentucky. Most states will get another round of “fiscal recovery funds” — part of President Biden’s $1.9 trillion American Rescue Plan — next month.

But in Washington, Mr. Biden is out of money to pay for the most basic means of protecting people during the pandemic — medications, vaccines, testing and reimbursement for care. Republicans have refused to sign off on new spending, citing the state recovery funds as an example of money that could be repurposed for urgent national priorities.

“These states are awash in money — everybody from Kentucky to California,” said Scott Jennings, a former aide to Senator Mitch McConnell of Kentucky, the Republican leader. “People are like: ‘We’ve printed all this money; we’ve sent it out. These states have these massive surpluses, and now you need more?’”

Republicans were never fans of Mr. Biden’s rescue plan, which Democrats muscled through Congress without their support. Despite the many ways it is benefiting his state, Mr. McConnell once called it a “multitrillion-dollar, nontargeted Band-Aid” that would dump “another huge mountain of debt on our grandkids.”

On Capitol Hill on Thursday, a day after Mr. Biden made a public appeal to Congress for more money, Senate Republicans and Democrats were nearing a deal on a $10 billion emergency aid package — less than half of Mr. Biden’s initial request. But they had not resolved crucial differences over the size and how to pay for it. Republicans want to use unspent money already approved by Congress, but the parties have been unable to agree on which programs should be tapped.

Since the outset of the pandemic, the Trump and Biden administrations have injected $5 trillion into the American economy, including the rescue plan. With midterm elections approaching, the gush of federal stimulus spending will draw even greater scrutiny as Republicans accuse Democrats of wasting funds and fueling inflation, and demand a precise accounting of how the money has been spent.

David Adkins, the executive director and chief executive of the Council of State Governments, said such questions were inevitable now that policymakers could catch their collective breath.

“We have to lean into the notion that states are laboratories of democracy,” Mr. Adkins said. “Some of these things will fail; some of this money will not be spent well. But that is the nature of trying to navigate disruptive times.”

The rescue plan set aside $195 billion to help states recover from the economic and health effects of the pandemic. When Mr. Biden made his initial aid request, senior lawmakers in both parties negotiated a plan to pay for it partly by taking back $7 billion from states, as part of a $1.5 trillion spending bill.

Governors and rank-and-file Democrats balked, saying that to do so would disproportionately hurt the 31 states that have not yet gotten all their rescue funds, and the deal fell apart. Now it appears the state funds will be spared, though the fracas has cast a sharp spotlight on how the fiscal recovery funds are being spent.

“I was never for giving this money to the states, but I was always of the belief that once you gave it to them, politics would not allow you to get it back,” Senator Roy Blunt of Missouri, the top Republican on the subcommittee that controls health spending, said in a recent interview.

All told, the White House says 93 percent of the American Rescue Plan dollars that are currently available have been “legally obligated,” meaning they have either already been spent or are committed to being spent.

Most states have either started spending their fiscal recovery funds, or have plans to do so. A recent analysis by the Center on Budget and Policy Priorities found that while most states are still developing budgets for the upcoming fiscal year, states have already budgeted 78 percent of their fiscal recovery fund allocation.

Kentucky, where Mr. Beshear, a Democrat, is promoting record job growth and economic boom times, ended 2021 with a record $1.1 billion surplus, and another surplus is expected this year. The state has already received $1.1 billion in federal funds and expects another $1 billion in May. It is spending the money on broadband, bolstering tourism and shoring up the unemployment insurance fund as well as coronavirus testing, in addition to water improvements.

“These dollars are too important and too transformational to get caught up in a partisan fight,” Mr. Beshear said in an interview, adding: “These are dollars that are helping us as we emerge from Covid. We’ve got a choice to limp out of the pandemic or sprint out of the pandemic, and cutting off this aid only hurts the people that need it.”

Congress specified four broad purposes for the money: to respond to the pandemic’s health and economic impacts; to provide bonus pay to essential workers; to prevent cuts in public services; and to invest in sewer, water or broadband infrastructure. But states can also use the funds to replace lost revenues, which gives them great flexibility in spending the money.

Arkansas, for instance, has awarded $374,000 to a rape crisis center; $6.3 million to the Arkansas Coalition Against Sexual Assault; and another $6.3 million to the Arkansas Alliance of Boys & Girls Clubs. But the bulk of the money has gone toward improving broadband access and addressing the needs of the health care system.

“The Omicron variant came in, cases skyrocketed, hospitals filled up and so we had to utilize a significant amount of our ARPA money for expanding hospital space, home testing and other public health response,” said Gov. Asa Hutchinson, a Republican, using the acronym for the rescue plan. “So that’s obviously the first responsibility, and then we looked at these other needs.”

Other states are using the money in ways that are only tangentially related to Covid-19, but that are permissible under guidelines issued by the Treasury Department.

Alabama devoted $400 million of its allocation, or roughly one-fifth, to building two new prisons, despite a public outcry from advocates for racial justice and civil liberties. Florida devoted $2 billion, nearly one-quarter of its $8.8 billion allotment, to highway construction — a decision that has drawn criticism from the nonpartisan Florida Policy Institute.

“The intended purpose of the American Rescue Plan Act dollars was to ensure that individuals and communities could recover from the pandemic, and I think in many ways there were better uses for this money,” said Esteban Leonardo Santis, the group’s tax and revenue analyst.

Twenty states, including Kentucky, spent a total of $15 billion to build up their depleted unemployment insurance trust funds. Independent analysts say that is effectively a tax break for businesses, which otherwise may have had to make up for the lost revenues. But Mr. Beshear defended it, saying that Kentucky businesses stepped up during the pandemic. A local Toyota plant made face shields, and bourbon distillers manufactured hand sanitizer, he said.

The governor’s Twitter feed is rife with photos of big checks and smiling city and county officials; he is running for re-election in 2023.

“If there’s one thing a governor knows how to do, it’s drive around their state and hand out huge checks and cut big ribbons with oversized scissors,” Mr. Jennings said. “They’re like game show hosts out there.”

Experts say, and the White House acknowledges, that the fiscal recovery funds have helped create state budget surpluses. Gene B. Sperling, a senior adviser to the president who is overseeing the American Rescue Plan, said the surpluses were proof that Mr. Biden’s stimulus package was working — and this was no time to pare back.

“Ensuring that states and localities have a cushion for some pretty serious bumps in the road is smart policy,” Mr. Sperling said, “and a lesson learned from what happened after the Great Recession.”

But those surpluses are likely to be temporary, and how states are using them has played into the controversy over Covid relief funds. The Center on Budget and Policy Priorities says 14 states are using temporary budget surpluses “to call for costly and permanent tax cuts targeted more to wealthy people” — a move the center described as a “bad choice.”

Here in Frankfort, the state capital, Kentucky lawmakers in a hurry to wrap up their 2022 legislative session were working on pushing through a hefty income tax cut this week. But a proposal to use the state’s budget surplus to give Kentuckians a tax rebate of up to $500 seemed unlikely to pass, said its author, State Senator Chris McDaniel, the appropriations committee chairman.

Mr. McDaniel, a Republican, spent much of this week immersed in budget talks, including planning how to use Kentucky’s next tranche of fiscal recovery funds. Another $1 billion is coming, and despite some philosophical misgivings, he said he saw no reason not to spend it.

“I believe firmly that it was too much money that came down,” Mr. McDaniel said. “But I also believe that Kentuckians will bear the tax burden eventually, just like everyone else down the line, and I am not going to disadvantage future Kentuckians out of a point of philosophical pride.”

Emily Cochrane contributed reporting from Washington.