Is it permissible to move to another mosque for prayer if the imam of the original mosque hires someone to deputize for him in leading the prayer, and that person is not competent?

What the Imam does by falling short of his Imamate is a betrayal of the responsibility entrusted to him. Appointing a substitute does not absolve him, especially if the substitute is incompetent or unauthorized. It is permissible to abandon prayer in this mosque and pray in another, provided that this does not lead to the abandonment of the neighborhood mosque.