Coun Box, who has led the local authority since 1998, has faced calls from within his own Labour group to stand down after Ofsted rated Wakefield's children's services inadequate.
He survived a vote of 'no confidence' at a party meeting in September, but four councillors then resigned from senior positions in protest at his leadership.
Sources within the Labour group say they remain intent on calling for Coun Box to stand down, despite chief whip Coun Richard Forster insisting that the majority of councillors support the leader.
Following a Cabinet meeting at Ossett Town Hall on Monday morning, members of the public were allowed to ask Coun Box and other senior councillors questions.
Asked whether or not he still commanded support within his own group and how long he intended to remain in charge, Coun Box simply responded: "Next question".
On Thursday evening, members of the Pontefract branch of the Labour Party formally called for another vote of 'no confidence' to be taken at the next meeting of party councillors.