Councillors were told the decision to bring forward the project earlier than expected, ground conditions and changes to a bridge design had contributed to a rise in the cost of the Wakefield Eastern Relief Road.
Network Rail’s decision to demand £1m for access to its land as part of the construction also added further costs.
The West Yorkshire Combined Authority originally allocated £32m to pay for the scheme.
But members of the authority’s investment committee heard the project could now cost £37.6m although discussions are continuing with the contractor, Network Rail and Wakefield Council.
Coun Box, who is also chairman of the combined authority’s investment committee, told the meeting: “It is going to create thousands of jobs, it is going to create over 2,000 new houses and it has already had a huge impact on air quality.”
He added: “Of course any lessons from this that need to be learned will be.”
The meeting also saw the committee approve £9m to support the development of a range of infrastructure schemes.
Councillors agreed to spend £570,000 to draw up detailed proposals to improve congestion at Ainley top and £317,000 to develop a scheme for improvements at Castleford station.
They also approved £4m should be contributed towards the cost of a project to use heat generated by the Aire Valley waste centre to heat local buildings.