“Since taking office, Governor Brownback has increased state funding to schools every year, investing more than $4 billion — approximately half of the state’s budget — in K-12 funding,” said Eileen Hawley, a spokeswoman for Mr. Brownback. “At the same time, he has returned more local control to those closest to the classroom — teachers and parents — so they have more direct control over how funds are spent to benefit students.”

District officials say, however, that the funding has not kept pace with a rise in fixed costs. The Shawnee Mission School District, for instance, faces an increase of $1 million in a year for bus services. Other expenses, even for school supplies, have grown in many districts, frustrating officials who say they need more help from the state to cover costs.

Under Mr. Brownback, who took office in 2011, state aid per pupil dropped to $3,800 from $4,400, according to the Kansas branch of the National Education Association. That reduction has come even as enrollment and the cost of health insurance have increased for many districts, said Mark Tallman of the Kansas Association of School Boards.

“Since the Great Recession, when you adjust for inflation, total school funding in Kansas has been basically flat,” he said.

Tensions flared when the State Supreme Court first ruled in February that unless poorer districts begin receiving more money by the end of June, the state’s public schools could be shut down. Mr. Brownback signed a bill in April intending to fix the education funding formula. But the court on Friday said the measure had not solved the problem and reiterated the deadline.

What happens next is not clear. Schools in Kansas are generally finished for this academic year and Wednesday was set to be the Legislature’s last day, though it could return for a special session. Conservative lawmakers have denounced activist judges and Mr. Brownback has accused the court of “political brinksmanship.” But the governor had hinted that he might cut funding to Medicaid and higher education if the Supreme Court ordered the state to allocate more money for schools.