Is it permissible for the Imam to pay the difference from his own money to a good contractor without the knowledge of the original donor, if the donor had brought a bad contractor and the construction of the mosque is threatened by poor workmanship?

It is not permissible to agree with a contractor to build a mosque if the donor has rejected them, because an agent is bound by the permission of their principal. If the donor refuses to deal with a specific contractor, it is not permissible to deal with that contractor; otherwise, the agent would be transgressing. This action involves deceit towards the donor, who intended to build the mosque alone. If the chosen contractor is unsuitable, the obligation is to offer advice and clarification, and the donor has the option to make their own decision.