It was revealed last week that the local authority has had to pay £17,500 a week to a building contractor for six months, despite the firm having been unable to do any work in that time.
McConnell were tasked with fixing the roof at Sandal Magna Community Academy in 2019, but work halted last summer after serious rot was found in the structure of the school's main teaching block.
A clause in the contract, which the council says is "standard" across industry, means that the local authority has had to pay McConnell £430,000 since, despite the builders having been off site.
On Tuesday, councillors are expected to commit more money to the repairs which, if resumed shortly, would allow the scheme to be completed by October.
Sandal Magna's pupils, who've suffered from repeated closures of the school because of the leaky roof in recent years, are currently being taught in temporary classrooms because of the issues.
Beate Wagner, Wakefield Council’s corporate director for children and young people, said: “The health and safety of both students and staff at Sandal Magna Community Academy has to be our first priority, which means we need to provide a fit for purpose teaching and learning environment and significantly reduce the risk of future regular school closures.
"The works necessary could not have been foreseen prior to work beginning and although they create significant additional cost for the council, the industry standard contract we have in place with the contractor is there to ensure they are not disadvantaged by an enforced pause in construction.
"Approval of additional funding will enable all remaining defects to be addressed and for the school building then to be in a position to be handed over to the academy."
Academies are normally funded directly from the Department of Education and their own private sponsors.
But Wakefield Council was told it had to pay for the damage to the roof, because the problems originated from when the school's new building was constructed in 2010, three years before Sandal Magna left the local authority and gained academy status.
A Cabinet report published last week indicated the council may pursue legal action against the firm responsible for the new building 11 years ago, despite having previously ruled it out.
The report said that the authority was seeking "internal legal advice again "given recent surveys indicate a flawed design as the reason for the defective roof."
Local Democracy Reporting Service