City Council approved a $513 million budget on a 5-3 vote at its meeting Wednesday night.

The three-vote opposition, led by Councilman Mike Duman, favored finding a way to provide an additional $1.1 million to the schools to replace heating, ventilation and air conditioning equipment in response to a request from School Board Chairman Mike Debranski.

“My personal opinion is that I think now is a good time to start on our new relationship, and there has to be a trust factor,” Duman said.

Duman proposed allocating the money from the capital reserve fund and replenishing it later with any year-end surplus that might be available.

Councilmen Roger Fawcett and Lue Ward, in their first budget vote since being elected, sided with Duman.

“We need to try to find a way to get it done,” Fawcett said, noting that a new system would likely result in utility savings. “I fully support trying to get some relief to them.”

After other council members noted that the systems could be replaced if they failed, Ward chimed in, “It’s 100 degrees, no windows, and some kid has got asthma — are you going to respond then?”

Mayor Linda T. Johnson said she did not approve of using the money in that fund for that purpose, noting that other needs may come up. Schoolchildren could be sent home in an emergency, she added.

“If it broke tomorrow, that’s an emergency,” she said. “That’s a whole different world.”

Councilman Curtis Milteer added: “If they’re not broke, don’t fix them.”

After a failed, 3-5 vote on amending the proposed budget to allocate the money, a motion to pass the budget as presented passed on the flipped vote.

In comments later in the meeting, Debranski said he was “dismayed” at the decision but appreciated the consideration.

The school system has in the past faced lawsuits alleging excessive levels of mold in school buildings that sickened students and staff.

The approved budget hikes the real estate tax rate by 6 cents per $100 of assessed value. It provides $3 million over last year’s funding level to Suffolk Public Schools, which was $6 million short of the division’s request.

Water and sewer rates are set to rise by about 10 percent, to $7.84 per 100 cubic feet for water and $5.82 per ccf for sewer.

The budget freezes eight vacant positions and cuts money for things like equipment replacements and legislative services, even though general fund spending rose 3 percent to $179.3 million.

Two capital projects were delayed to save money, but other capital projects will see the city take on nearly $34 million in new debt.

Later in the meeting, City Council voted to schedule a joint meeting with the School Board for August to begin a new level of year-around communication between the two bodies.