The governor was firm: Nebraska would reject the new federal money for summer meals. The state already fed a small number of children when schools closed. He would not sign on to a program to provide all families that received free or cut-rate school meals with cards to buy groceries during the summer.

“I don’t believe in welfare,” the governor, Jim Pillen, a Republican, said in December.

A group of low-income youths, in a face-to-face meeting, urged him to reconsider. One told him she had eaten less when schools were out. Another criticized the meals at the existing feeding sites and held a crustless prepackaged sandwich to argue that electronic benefit cards from the new federal program would offer better food and more choice.

“Sometimes money isn’t the solution,” the governor replied.

A week later, Mr. Pillen made a U-turn the size of a Nebraska cornfield, approving the cards and praising the young people for speaking out.

“This isn’t about me winning,” he said. “This is about coming to the conclusion of what is best for our kids.”

Mr. Pillen’s extraordinary reversal shows the conflicts shaping red-state views of federal aid: needs beckon, but suspicions run high of the Biden administration and programs that critics call handouts.

The new $2.5 billion program, known as Summer EBT, passed Congress with bipartisan support, and every Democratic governor will distribute the grocery cards this summer. But Republican governors are split, with 14 in, 13 out and no consensus on what constitutes conservative principle.

One red-state governor (Sarah Huckabee Sanders of Arkansas) hailed the cards as an answer to a disturbing problem. Another (Kim Reynolds of Iowa) warned that they might increase obesity. Some Republicans dismissed the program as obsolete pandemic aid. Some balked at the modest state matching costs. Others hinted they might join after taking more time to prepare.

The program will provide families about $40 a month for every child who receives free or reduced-price meals at school —$120 for the summer. The red-state refusals will keep aid from about 10 million children, about a third of those potentially eligible nationwide.

The rejection of federal aid has parallels to the bitter fights over the 2010 Affordable Care Act. Ten states, mostly Southern and low income, decline to run an expanded Medicaid program largely financed by Washington.

Still, some analysts find the rejection of the grocery cards surprising. Summer EBT is much cheaper for states than Medicaid, it passed Congress with Republican support and it grew from a pilot program widely deemed successful. Plus, it targets children.

“It should be less controversial than it’s been,” said Elaine Waxman, a hunger expert at the Urban Institute, a Washington research group.

The outcome illuminates the arbitrary nature of the American safety net, which prioritizes local control. North Dakota and North Carolina are in; South Dakota and South Carolina are out. Children can get aid in Tulsa but not in Oklahoma City, as state and tribal governments clash. In the impoverished Mississippi Delta, eligibility depends on which side of the Mississippi River a child lives.

As with Medicaid, poor states are especially resistant, though the federal government bears most of the cost. Of the 10 states with the highest levels of children’s food insecurity, five rejected Summer EBT: Louisiana, Oklahoma, Mississippi, Alabama and Texas.

Like the school lunch program, it serves families up to 185 percent of the poverty line, meaning a family of three would qualify with an income of about $45,500 or less.

The initial school meal program faced resistance, too. Congress created it in 1946, partly from fear that poor nutrition weakened military recruits. But opponents saw free meals as socialism, and Southern states demanded assurance that federal aid would not undermine segregation.

More than a decade later, only half of schools ran the program, Susan Levine, a historian at the University of Illinois Chicago, noted in her 2008 book “School Lunch Politics: The Surprising History of America’s Favorite Welfare Program.”

A separate Summer Food Service Program followed in 1968. But it offers meals at limited sites, which some families cannot reach, and serves only about 15 percent of children fed during the school year.

Some critics see the new program as an extension of pandemic aid. (A similar effort, Pandemic EBT, distributed grocery cards when the coronavirus closed schools.) But Summer EBT, having started experimentally in 2011, long predates the pandemic. Evaluators found that even benefits as low as $30 a month cut “the most severe food insecurity among children by one-third.”

Drawing on those results, Congress in 2022 established the program nationwide. In exchange for the federal benefits, states pay half the administrative costs. Perhaps sensing some might resist, a Republican backer, Senator John Boozman of Arkansas, said in a promotional video, “We’re counting on you to put these new tools into action.”

His home-state governor, Ms. Sanders, did. As a White House press secretary under President Donald J. Trump, Ms. Sanders does not want for conservative credentials, but she celebrated the federal aid.

“Making sure no Arkansan goes hungry, especially children, is a top concern for my administration,” she said in a news release. Arkansas officials estimate the program will cost the state about $3 million and deliver $45 million in benefits.

Iowa rejected the program with equal verve. In forgoing about $29 million in federal benefits, Governor Reynolds called the program “not sustainable” and criticized the lack of constraints on which food parents can buy. “An EBT card does nothing to promote nutrition at a time when childhood obesity has become an epidemic,” she said.

The pilot program found the opposite: EBT cards “increased consumption of fruits and vegetables,” evaluators wrote, and lowered the consumption of soft drinks.

More than half of the children whom Republican governors have excluded from aid live in Texas and Florida. Both states have noted the program’s administrative complexity: Schools often lack current student addresses or the technology to share data easily with agencies that issue EBT cards. But neither has ruled out future participation.

The Biden administration, seeking to protect the program from a partisan gloss, has generally not criticized states that refused the aid.

“A number of the nonparticipating states have told us they were challenged by the timeline and hope to implement the program next year,” said Stacy Dean, the deputy under secretary of agriculture.

Some Republicans, in rejecting the aid, found critics in their own ranks. After Gov. Henry McMaster of South Carolina dismissed Summer EBT as a duplicative “entitlement,” State Senator Katrina Shealy, a fellow Republican, wrote a column with a Democratic colleague warning that “hunger does not stop during summer break.”

In an interview, Ms. Shealy said the state should not reject $65 million “just because Biden is president,” and perhaps just partly tongue-in-cheek wrapped her plea in Trumpian bunting: “Everyone wants to say, ‘America First’ — well, let’s feed our children first.”

Oklahoma initially said it rejected the program because federal officials had not finalized the rules. But responding to critics, Gov. Kevin Stitt, a Republican, sharpened his attack, calling Summer EBT a duplicative “Biden administration program” that would “cause more bureaucracy for families.”

Tribal governments, which have influence over large parts of the state, stepped in. Already feuding with Mr. Stitt, they promised to distribute cards to all eligible families on their land, regardless of tribal status, while bearing the $3 million administrative cost. The five participating tribes will cover nearly 40 percent of Oklahoma’s eligible children, most of them not Native American.

“I remain dumbfounded that the governor of Oklahoma would turn down federal tax dollars to help feed low-income children,” said Chuck Hoskin Jr., the principal chief of the Cherokee Nation.

In Nebraska, Governor Pillen was an unlikely candidate to support a new poverty program. A wealthy pork processor, he ran on vows to fight critical race theory, resist “the federal government’s invasion” and “keep the socialist agenda out of our state.”

The existing meal sites were preferable to Summer EBT, he said, because they let children socialize and allowed staff members to check their well being. But many families lack the time and transportation to get children to the sites, especially in rural areas.

Historically, the number of children Nebraska feeds during the summer is only about 7 percent of those fed in school, one of the lowest ratios in the country, according to the Food Research & Action Center, a Washington advocacy group. By contrast, Summer EBT would reach nearly every family eligible for a subsidized school meal.

After the governor rejected the program, thousands of Nebraskans signed a protest petition, and 19 members of the unicameral legislature backed a bill to force the state’s participation. They included Senator Ray Aguilar, a senior Republican, who said in an interview that the program reflected conservative values because “kids need to eat.”

The state’s fiscal analysts estimated that the program would cost about $360,000 a year and bring $18 million in benefits.

Megan Young, 25, does not follow politics but heard about the dispute. Relying on school meals growing up, she ate less in the summer and watched her mother go hungry. Food insecurity, she said, deepened her mother’s depression, which sent her into foster care. “I was shocked,” she said, to hear the governor call EBT cards “welfare.”

Ms. Young was in a program that let disadvantaged teenagers and young adults lobby the governor on an issue of their choice. Her group chose Summer EBT.

Standing before Mr. Pillen in the State Supreme Court, Matthew Floyd, 18, said federal cash would help the economy. Lexie Simonsen, 18, brought a brown-bag lunch to argue that the meal-site fare was meager and unappealing.

Ms. Young spoke in the most personal terms, explaining that “my mother would go without or very little so that we could eat.” She did not tell the governor about her journey into foster care and homelessness, for fear he might find such hardship implausible.

The presentation lasted eight minutes. The governor seemed unpersuaded. He doubted the group’s estimate of how many children would benefit from the cards. He insisted that the summer sites met children’s needs. “I’m not asking you to agree with me,” he said.

The group left dejected.

Ms. Simonsen was in study hall a week later when she learned that the governor had reversed himself and announced Nebraska would send out the EBT cards. He cited several youth groups, including hers, for altering his view.

It was unusual, she said, for powerful men to change their minds, and she credited Mr. Pillen, a former college football star, for not making the issue a contest he had to win.

“The fact that he listened and said Nebraska can do better — that blows my mind,” she said.