The council has been paying the firm McConnell £17,500 a week since October, after work to fix the leaky roof at Sandal Magna Community Academy stopped.
The revelation led to the council's leader of the Opposition, Nadeem Ahmed, to call for an inquiry.
At a Cabinet meeting on Tuesday morning, council leader Denise Jeffery said the authority was "accountable to the public" and promised that a report on the issue would be published at the earliest opportunity.
McConnell had been appointed in late 2019 to repair Sandal Magna's defective roof, which has been blamed for causing repeated closures at the school in recent years.
But when rot was found in the structure of the school's main teaching block the following September, workmen were forced off the site while further investigations took place.
As a result of a clause in the McConnell's contract with the council, which the authority said is standard across industry, the firm claimed £430,000.
The council said on Friday that the issues "could not have been foreseen", but speaking before Tuesday's Cabinet meeting, Coun Ahmed said: "This is money from Wakefield district taxpayers and the measly explanation from Wakefield Council is simply not good enough.
"Residents of the district deserve better than this and certainly deserve answers to why the council have spent nearly half a million pounds on a leaky roof. This is gross negligence and someone has to take responsibility for this.
"I will be calling for a full independent inquiry into this matter and I demand the report and its findings should be presented to full council for discussion and not just the Cabinet."
The council is also seeking legal advice over whether or not to take action against the firm responsible for building Sandal Magna's new school building in 2010, after a report described the roof as "defective" because of a "flawed design".
Coun Ahmed added: "The council should pursue all legal avenues with the construction firm with a view to seeking full compensation for the taxpayers of Wakefield.
"Sadly, this is yet another example in a long list of Labour's incompetence with taxpayer's money."
The council's sole Liberal Democrat councillor, Tom Gordon, tweeted on Friday: "It’s scandalous that almost half-a-million pounds has been given to contractors in return for absolutely nothing. Labour just can’t be trusted with your money."
Coun Jeffery addressed the issue directly at Cabinet on Tuesday as councillors were asked to commit more money to repair the additional defects found within the building.
She said: "Yes we've a duty to look after the children of our district and to provide the best quality schools that we can.
"But we also have a duty to the public, so I'm asking for an inquiry into what's happened with the funding, so that we are accountable.
"We are accountable to the public. It's their money and we need to have a report into what's happened, which I shall be bringing to council at the earliest opportunity."
Councillors agreed to commit an extra £1.7m of funding to ensure that the repairs are completed, taking the total cost of the scheme past £3m.
The job is now expected to finish in October, after which Sandal Magna's pupils will no longer have to be taught in temporary classrooms.
Some of the school's students have even been having lessons in a nearby council building because of a lack of available space on the site.
Had the council chosen to terminate the contract and abandon the repairs, it would have incurred a similar immediate cost.
Cabinet member Jacquie Speight defended the authority, saying there was "little alternative" but to ensure that the work was completed for the benefit of Sandal Magna's pupils.
She told the meeting: "I realise this is a very difficult issue and the leader has explained that steps will be taken to look at some of the issues surrounding it.
"But I do want to say that it's really important that as a council we support our children.
"Any disruption to children's education, the cost of that is incalculable."
Local Democracy Reporting Service