There are both mimosa trees and mimosa ground cover which I explain in this text. The trees can spread seeds quite thickly so that dozens of little trees will get started. They are easy to grow in the moderate to warm climates. Most people just mow them off when they cut their grass.
I realize that some people are especially particular about their lawns. I do not know what would be the best actions to take to get rid of these small spouts you describe.
My best advice is to contact your local or regional agricultural county agent. There is no charge as our taxes pay for this service whether anyone uses it or not, so why not use it? They should have good advice on how to proceed with the problem you describe.
Locate the phone number and physical location of your specific county agent by putting the words, "county agricultural extension office," or "county agricultural agent," in the Google search box. Instead of saying "county," place the name of the county you live in or the county where the mimosa problem is, in front of the word county. For example, "Madison County agricultural extension office," or Columbia County agricultural agent/extension office."