No power. No authority. No influence other than the freedom to express their views and see who else is persuaded by their arguments.
The problem is that government has to have the ability to enforce its laws, so it must be able to compel obedience. The only way to preserve any individual liberty is to limit the scope of the government's laws and actions.
Religion, on the other hand, has an unlimited scope. It addresses every aspect of life. Therefore, it must be kept separate from the compulsive power of government. Otherwise, the temptation to use that power is overwhelming.
Historically, whenever the two have been allowed to combine, the result has been oppressive government and corruption of the religion.