The think tank has initiated a campaign calling for council scrutiny committees to be allowed to look at any combined authority plans before they are agreed by Whitehall and local authority leaders.
The campaign also calls for all devolved areas to have a governance framework to show how the combined authority will bring the public into the decision-making process.
Lord Kerslake, CfPS chair, said: “An elected mayor does not itself guarantee that these new powers will be used in a fair, transparent and democratic way.”