School budgets passed by large margins in most school districts stretching from Long Island's eastern tip to the Nassau-Queens border, though results were incomplete Tuesday night because of complaints of voting irregularities in Hempstead.

Spending plans were reported passed in 122 out of 124 districts in Nassau and Suffolk counties, with one defeat reported in Patchogue-Medford, where a parent group's petition for extended busing put the district's tax increase over the state-imposed cap limit.

In Hempstead, a State Supreme Court justice ordered machine ballots sealed after complaints surfaced over absentee ballots there.

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In districts where budgets won approval, the margins of victory were among the highest on record: Bay Shore's $146.8 million budget passed 1,700 to 608; Wantagh's $75.7 million plan, by 1,281 to 271; Jericho's $121.1 million budget, 659 to 149.

"I think it's great -- it seems like the community is satisfied," said Greg Nardone, president of Bay Shore's school board.

Hank Grishman, superintendent of Jericho schools and a former president of the New York State Council of School Superintendents, noted that voting was generally light -- a phenomenon statewide in recent years.

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"But results were overwhelmingly positive," Grishman added. "I think the voting public appreciated the conservative tax levy increases."

Margins of victory in many school board races were far narrower, and some incumbent board trustees were ousted, suggesting that financial stability was offset to a degree by political instability.

This year's proposed spending hikes, in budgets totaling more than $11.9 billion in Nassau and Suffolk counties, are at their lowest point on record, with the Islandwide increase averaging 1.69 percent. Tax increases also are at a near-record average low of 1.71 percent.

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Gov. Andrew M. Cuomo's office has promised to start mailing out tax-rebate checks in the fall to residents of districts that keep within tax-cap guidelines and meet the state's new "efficiency" rules for cutting costs. Rebates averaged $230 per household in Nassau County last year and $113 in Suffolk, and are expected to more than double this year.

Only two Long Island districts -- East Meadow and Patchogue-Medford -- had ballot propositions that would cause spending to exceed their respective state tax caps during the 2015-16 school year. In each of the districts, a supermajority of 60 percent of those voting was required for passage of spending plans and attendant propositions.

East Meadow's proposition, approved by the local school board, funds full-day kindergarten. Returns Tuesday night showed the budget and proposition had passed handily.

Patchogue-Medford's proposal, initiated by parents whose children attend Catholic schools, would extend bus transportation for those students and others in private schools. That plan was defeated, in a repeat of what happened last year.

Budgets passed in three other districts -- Elwood, Mount Sinai and Northport-East Northport -- that had proposed to open full-day kindergartens in the fall. All three kept within tax caps. Floral Park-Bellerose and Harborfields are the only remaining districts on the Island without full-day programs, though Floral Park-Bellerose moved to an extended four-hour day last year.

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"It's a great thing," said Mount Sinai board trustee Lynn Jordan.

Confusion in Hempstead

Meanwhile, the Hempstead school district -- where allegations of irregularities led to a special election last fall -- appeared headed for another protracted electoral fight. Confusion reigned at the Alverta B. Gray Schultz Middle School polling site, where voting ended at 9 p.m.

District officials stopped the count of ballots after an attorney for candidate Maribel Toure raised questions about absentee ballots, said Scott Kershaw, an attorney for the Hempstead schools.

With passage of many school budgets virtually assured, attention turned to many of the more than 200 contested school board races across the region.

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Organizers of the Long Island Opt-Out group -- mostly parents who pulled their children in grades three through eight out of the state's standardized English Language Arts and math tests -- had endorsed at least 75 candidates up for election in districts stretching from Westhampton Beach to Valley Stream.

As of late Tuesday night, 53 candidates endorsed by the grassroots group had won board seats, while 21 were defeated.

Local teachers unions also supported candidates in dozens of districts, and taxpayer groups backed rival slates in some of those same communities.

Bayport-Blue Point contest

Two board seats in Bayport-Blue Point were hotly contested.

Rebecca Campbell, an attorney and the board's incumbent president, was challenged by Christopher Richardt, a school principal in Hampton Bays. Two other candidates, Gina Murphy, a Brentwood teacher, and John Vazquez, an American Express director of business development, were vying for a second slot.

Both Richardt and Murphy were endorsed by Bayport-Blue Point's teachers union, and both were elected Tuesday.

Many voters casting ballots Tuesday at Bayport-Blue Point High School said they turned out primarily to support the district's $69.5 million budget. Some said they also were drawn by the board races.

In East Meadow, a stream of mothers entered the polling area at George McVey Elementary School to cast ballots on Proposition 1, the $192.8 million regular budget, and for Proposition 2, providing an extra $2.9 million for full-day kindergarten.

"We really want full-day kindergarten to pass," said Sharan Sokhey, 33. Her older son, 5, will enter that grade level in the fall. "I've talked to other parents, and it seems once children are in first grade, more is expected of them because of Common Core."

With Jo Napolitano

and Joie Tyrrell

CORRECTION: An earlier version of this story misstated where Christopher Richardt was principal.