Voters on Tuesday will get a second chance to decide on proposed school budgets in Tuckahoe, which seeks to override its state tax cap, and Elwood, which is keeping within cap limits.

The two Long Island districts are among nine across the state that have scheduled budget revotes, according to the New York State School Boards Association. Tuckahoe is among four systems statewide attempting for a second time to overcome cap restrictions on increases in property taxes for the 2016-17 school year.

A total of 676 districts statewide and 124 in Nassau and Suffolk counties held initial budget votes last month.

Under state law, school systems intending to bypass cap limits must garner majorities of 60 percent or better to get budgets approved. Districts failing to win such majorities on two consecutive ballots face tax freezes the following year.

School officials in both Tuckahoe and Elwood voiced optimism over the expected outcome of the revotes. Past voting patterns indicated that Tuckahoe faced the bigger challenge in rustling up enough extra “yes” votes to pass its revised spending plan.

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Last month, Tuckahoe residents voted 146-145 in favor of the district’s $19,533,000 plan, which carried a 2.3 percent spending hike. Under the 60 percent rule, the tiny East End district would have had to increase the number of “yes” votes by nearly 50 percent to succeed.

The defeat came despite attempts by the district to economize by eliminating a summer-school program that usually enrolls about 100 children.

Allan Gerstenlauer, who serves as Tuckahoe’s part-time superintendent, acknowledged that a second override attempt entailed some risk. A tax freeze next year would force the district to draw down cash reserves, leaving it with little financial cushion for the future, he said.

The superintendent contended, nonetheless, that his district did the right thing in declining to cut student services any further. East End districts in recent years have experienced better luck in overriding caps than districts farther west, in large part because their taxes are lower.

Tuckahoe, like other districts, has struggled this year with the tightest state tax restrictions imposed since the cap system took effect in 2012. Baseline cap levels are calculated according to inflation rates, which are running this year at unusually low levels.

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“I am confident that this is the right course of action for the district, or I would not be recommending it,” Gerstenlauer said. “Given the hard work by our school board and what this means to our community and out students, I am very confident that voters will support the budget.”

Tuckahoe enrolls 350 students in preschool through grade eight in its single school building, and sends older students to nearby Southampton High School on a tuition basis.

For the revote, Tuckahoe has come back with a lower $19,293,000 budget, representing a 1.03 percent spending increase. To pare expenses, the district has dropped plans to hire an additional security guard.

In addition, Tuckahoe will save an estimated $244,000 annually, because it negotiated a five-year tuition contract with neighboring Southampton. The district previously contracted with Southampton on a year-to-year basis and will get a discount under the extended agreement.

The revamped budget calls for a 1.75 percent increase in total tax collections, known as a levy, for the 2016-17 school year. The tax figure is less than the 2.48 percent raise projected in the first round of voting, but well beyond the district’s allowable 0.38 percent cap.

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Elwood, like Tuckahoe, tried to override its tax cap in the first round of voting last month and failed. Its margin of defeat was smaller, however.

The initial vote on Elwood’s $59,891,409 budget was 1,132 “yes” to 805 “no”. The 58.4 percent majority was just short of the 60 percent needed.

Nonetheless, the district has decided to keep within its cap for the second round of voting.

“The board got the message,” said Andrew Kaplan, the Elwood board president. “I’m really pleased that we’re able to present a budget that’s within the tax cap and that doesn’t make cuts directly in classroom programs.”

Elwood is a mid-sized district in Huntington Town enrolling about 2,270 students.

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For round two on Tuesday, Elwood has trimmed its spending request for next year to $59,458,599 — a 0.04 percent decrease from what the district appropriated this year. The measure requires only a bare majority to pass.

Cuts will include $150,000 in administrative costs and $15,000 in teacher training.

In addition, Elwood administrators said they will draw down further on cash reserves, even though they added that this could aggravate problems that recently prompted the state Comptroller’s Office to rate the district as “susceptible to fiscal stress.”

Elwood projected its tax-levy increase next year at 0.657 percent. That’s just within the cap, and lower than the 1.88 percent hike proposed in the first budget.

As in Tuckahoe, local officials said Elwood’s lower budget will preserve all curricular and extracurricular activities, while preserving class sizes at existing levels.