if a church has a performance license does the writer of those music make money from it?

Yes they do

They pay a fee to get the license. The fee gets distributed among copyright holders.

In most cases, the original writer has sold the copyright to a larger label or studio, who has paid them. The label/studio then worries about collecting the fees, and if the portion they get is "fair" for how often their songs are used, etc. Some licenses require reporting of the exact songs that actually are used, some don't.

Writers of music are not required to collect fees from everyone who licenses their songs. If a writer wants to license a church to use their music without charging them, they can simply donate the license to the church.

If they're not interested in donating to the church, they can collect the licensing fee from the church, just as they would from any other organization.

A license to perform said music?

Or just a license to put on performances?