I have never heard that expression. However clout means to be able influence others because of your reputation or power. So for example, If you said <he has a lot of clout in this town> you are expressing that he can influence many people and get whatever he wants within the town.

Clout chasing is when a particular person who doesn’t have any sort of social recognition tries for example befriend a famous person like, influencer, footballer etc. To be famous and to be known as their friend. Clout is a slang for “Fame”. So they’re chasing fame by doing an action for it.