Black locust? spreading
The roots are still alive and have a lot of energy stored in them. They will continue to put out shoots in an effort to survive for a while. If you keep mowing them down, it will eventually exhaust the reserves and that will kill the roots.
You could also try to speed up the process by spraying the shoots and leaves with a glyphosate herbicide, however this is a total vegetation killer and will kill anything it contacts, such as your lawn.
Easiest thing is to just keep mowing and let the situation resolve itself.
Yay! That’s easy... thanks, Ellen.