Prince William County supervisors preliminarily cut a substance abuse program for jail inmates, eliminated the Columbus Day holiday for county staff, and cut new police officers in future years from the county’s fiscal 2014 budget Tuesday, setting the stage for a final vote next week on a plan that would mean a small average tax increase for county residents.

The budget is scheduled to be finalized, and the real-estate tax rate set by supervisors, this Tuesday. But many of the tough decisions were made during this week’s “mark-up” session, where supervisors haggled over cuts big and small in a series of “straw poll” votes and negotiations.

Supervisor Martin E. Nohe (R-Coles) laid out proposals that called for an average flat tax bill for this year, or other proposals that would increase tax bills by about 2.5 percent on average.

The board settled on a budget that would mean an average 2.3 percent increase in residents’ real-estate tax bill on a non-binding 6-2 vote. Board Chairman Corey A. Stewart (R-At Large) and W.S. Covington III (R-Brentsville) voted against the proposal. Both had sought deeper cuts.

Under that scenario, the average county resident would pay $3,392 in real-estate taxes and the county’s operating fund would grow to $961.5 million, up from 914.1 million for fiscal 2013, according to Budget Director Michelle Casciato. The real-estate tax rate would be $1.181 per $100 assessed.

After much debate, supervisors declined to cut $700,000 from the county’s health department staff, which county officials said would affect the salaries of nearly 70 employees. They also kept funds for future libraries in Gainesville and Montclair, and fully funded the School Board’s most recent budget request of $888 million.

Supervisors disagreed most over cuts to the county health department and the two libraries. Supervisor Frank J. Principi (D-Woodbridge), who runs the Greater Prince William Community Health Center, a nonprofit health-care provider, said the cuts could be devastating to the county’s health clinic.

“That would be an across the board pay cut for a number of staff that would literally walk out the door,” he said.

The budget maintains funds for new voting machines, a new Central District Police Station and planned countywide technology improvements. It also keeps funds for additional School Resource Officers for security at county middle schools.

Because the county plans its budget over five-year periods, Nohe proposed changing the school funding formula slightly to give schools enough money for its operating fund proposal this year. The county usually automatically allocates 56.75 percent of its total operating budget to schools; that percentage was tweaked upward to 57.23 percent to fulfill the schools’ request.

Stewart disagreed with the change, saying that the county’s revenue split with the school system had worked well for the last 18 years. “You start changing it, and it’s going to start changing every year going forward. It’s a foolish thing to do for the long term,” he said.

While the board can add or cut programs and items every year, Nohe’s proposal showed what consistent funding levels would do for future expenses. A proposal for additional police officers to serve in a specialty tactical unit was tentatively cut, and a plan to add 15 police officers each year starting in fiscal 2015 was scaled back to an annual increase of 10 officers.

Under this budget proposal, the county should be able to maintain a plan that would add 27 new fire personnel annually starting in 2015, up from 13 per year, Nohe said.

Proposed cuts to health and social service programs had drawn scrutiny at the county’s public hearing on the budget recently, with many residents speaking against cutting services.

Cutting the Columbus Day holiday would save the county $450,000 per year. The board also made several smaller cuts, including eliminating a print newsletter and the county’s membership in the High Growth Coalition, a membership association that represents the interests of fast-growing localities in Virginia.

The budget is preliminary until supervisors formally adopt it, which they are expected to do on Tuesday.