Serco won a £70m contract in April 2015 to take over HR and finance support for Lincolnshire County Council.
Lincoln's Bishop King Primary School said about 30 staff were affected, with four receiving no wages at all.
Serco said it was "really sorry for the impact" the latest error had had on staff at the school.
Abi Tierney, delivery director for Serco, said there has been some "significant issues" with parts of the service.
She said the firm was "putting in a lot of work to deliver changes and make improvements", but  denied accusations the contract was too big for the company to handle.
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In May, Serco was fined more than £1m after failing to pay bills to suppliers and leaving staff without wages.
The fine came after a number of head teachers complained that errors by Serco meant staff were left unpaid.
Paula Colburn, business manager at Bishop King, said staff had been "left in tears" and put under unnecessary stress.
She said: "We can't keep working like this - I can't have it being detrimental to the staff's health."
In a statement, Lincolnshire County Council said: "Serco has committed to rectifying the payroll issues by the end of October, and we are satisfied that they are making good progress to achieve this."

The 20-year-old, who played 12 games for Rovers last term, will be with the League Two club until January.
Calder has yet to make his first-team debut for Villa, who were relegated to the Championship last term, but has been an unused substitute.
He played 11 games for Dundee in the Scottish Premiership in 2015-16, prior to his first spell with Doncaster.
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Ince, 25, who scored 38 goals in 115 games for Derby, has signed a three-year contract.
USA international Williams, 28, has agreed a two-year deal.
Chelsea's 20-year-old forward Kasey Palmer, who spent last season on loan with the Terriers, has agreed another season-long loan.
England Under-21 international Palmer has signed a new contract until 2021 with Premier League champions Chelsea.
Huddersfield manager David Wagner said of Ince: "He scores goals, creates them for others and is always a threat when he's on the pitch.
"I have spoken to Tom and he has a massive desire to come to Huddersfield Town and play in the Premier League.
"He has had a taste of the division for two short spells and he is desperate to show everyone that he belongs in the top division. He can be a big player for us."
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A company responsible for council payroll services has apologised after staff at a school were left without wages, or paid incorrectly.
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Doncaster Rovers have re-signed Aston Villa midfielder Riccardo Calder on a half-season loan.
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Huddersfield have signed Derby winger Tom Ince for an undisclosed fee and Reading midfielder Danny Williams after his contract with Reading expired.