Wakefield Council hired McConnell to fix the leaky school roof at Sandal Magna Community Academy in November 2019.
The work then halted last summer when rot was discovered in the school's main teaching block, which left the building unsafe to use and required further investigation. The builders have yet to return to the site.
However, a report on the issue has revealed a clause in the builder's contract means the council is still having to pay McConnell £17,500 a week.
Since October, the company, which is headquartered in Scotland but works across the UK, has claimed £430,000 through the clause.
The report, which will go before Wakefield's senior councillors next week, said a decision on whether to plough more cash into the roof repairs, or abandon the contract completely, needed to be made urgently.
It said: "After completing 50 per cent of the roofing project, in August 2020 the contractor moved to the main teaching block and uncovered more extensive timber decay issues.
"Removal of the roof revealed further significant structural concerns which required a wide range of invasive investigations during the winter of 2020.
"The contractor was unable to undertake any further work until the full extent of the work was assessed and an agreed action plan is in place.
"As is common, the contract has a clause for costs incurred by the contractor if the council suspends work for any reason."
After revealing the cost of the clause, the report added that, "A decision to enable the contractors to return to site as soon as possible or to terminate the contract is therefore critical."
Pupils at Sandal Magna, which is on Belle Vue Road in the city, have been taught in temporary classrooms since September, when Covid restrictions have allowed them to attend.
Some year groups have even been had lessons in a council office building on nearby Barnsley Road, because of the lack of available space.
The school has been hit by repeated closures because of the original problems with the roof, despite the premises having only been built in 2010.
Two contractors previously tasked with carrying out repairs went bust, before the school ran out of cash to pay a third firm which was halfway through the job.
The council has been asked to commit a further £1.7m to the repair scheme, which would take the total bill for the job past £3m.
The report said that should councillors approve the extra funding, the works will be able to continue and would be due for completion in October this year.
The council had originally ruled out taking legal action against the firm responsible for building the school in 2010, over fears any case would be too expensive and unsuccessful.
However, 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