Councillors heard a petition calling for recycling centres to be kept open and a total of 12 separate public questions on the issue.
Councillor Jillian Creasy then asked for the Green Party motion to be debated first. The Council’s constitution normally means this happens because of the level of interest in the public gallery.
Labour’s Julie Dore spoke against it and Labour councillors voted not to have the debate.
The Council’s Scrutiny Committee will now investigate the £300,000 profit said to be made by the sub-contractor and the involvement of A4E in providing unpaid labour to replace jobs.
Labour Councillors later voted down Green Party proposals for discussions with the Unions involved in the household waste sites strikes. They also opposed greater openness in broadcasting audio recording of council meetings.