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."
A company responsible for council payroll services has apologised after staff at a school were left without wages, or paid incorrectly.