Equally often, they engage in things that have the opposite effect. As an example, I would mention the opposition to marriage equality from both the Roman Catholic Church and the Church of Jesus Christ of Latter Day Saints. Opposition to a woman's right to choose is another example.
Money is used in some of these cases to influence churchmembers' opinions and even to influence others, through the purchase of advertising and other publicity. Similar campaigns are done for issues we might consider beneficial as well as for issues that we might consider to be detrimental.
Who decides what is beneficial and detrimental? That is the primary difficulty. So, we try to prohibit political activity by religious organizations, to the extent that we don't allow them to campaign for specific candidates.
No religious group should impose its particular viewpoint on anyone who is not a member of that group. That is the reason for the separation of church and state. It is a wise thing we do to keep the two separate.