With a record $220 billion state budget in the bag — and taxpayers on the hook — Gov. Kathy Hochul and the Legislature have unfinished business to conclude before the session ends June 2.

During the budget battle, the Legislature sidelined out key education items — extending mayoral control and raising the charter cap. And it balked at doing enough to fight rising crime.

Mayoral control expires June 30. Putting renewal off until the end-of-session “Big Ugly,” and so making it part of another round of bargaining with the gov, needlessly leaves families worried about the schools’ future and invites fine-print meddling to undermine the point of mayoral control, namely ensuring that voters can hold the mayor accountable for the system’s performance.

City schools are suffering a staggering 40% absenteeism rate, along with a serious drop in enrollment as many fled a system the last mayor seemed determined to make worse. This is no time to leave in doubt just who will be leading the system come July 1.

The Legislature should pass a stand-alone bill to extend mayoral control for at least four years as soon as lawmakers return from the Easter/Passover break.

Lifting the cap on public charter schools to allow more of these high-performing, quality programs to open in New York City would benefit thousands of students and families clamoring for better educational options. The waitlist for charter entry is 50,000 kids long; the Legislature needs to stop standing in the schoolhouse door.

Legislative leaders will fight tooth and nail against doing more to fix the Raise the Age and no-bail laws, but the need has only gotten more obvious since they agreed to a few small changes in the budget.

Tuesday’s subway attack and the ever-growing toll of kids lost to crime (which includes both victims and those seduced into gangs) make it plain that the streets will only keep growing more dangerous until the Legislature goes along with serious changes, including giving judges the discretion to send clearly dangerous offenders to jail. Lawmakers must also do more to reduce the impossible paperwork burden they’ve placed on prosecutors, which is forcing district attorneys across the state to drop cases against serious perps.

And the budget, while extending Kendra’s Law for five years, didn’t do enough on mental health, even as a failure to address the issue plainly contributed to the subway attack. For starters, get the job done on the measure to restore 200 state-run psychiatric beds that died late in budget negotiations. Beyond that, New York needs better laws that will make it easier to get those with dangerous mental illness off the streets and into mandatory treatment.

But lawmakers need to keep saying no to Hochul’s absurd and economy-killing ban on future oil and gas-hookups in new housing. Until enough alternative energy is plentiful and cheap, this is simply a cruel blow to lower-income New Yorkers.

The tax-exemption that encourages developers to build more affordable housing, 421-a, expires in June. It needs to be renewed in some practical form, or developers will stop building anything except luxury apartment buildings.

Right now, this fall’s election promises to bring a red Republican wave so powerful that it could swamp even deep-blue New York. If Albany doesn’t drop the extreme ideology and petty game-playing and start doing its job now, it’s going to be seeing a lot of new faces when the next Legislature convenes.