@Justin Johnson You are right in drawing up a new lease with her name on it along with the kid’s names / ages. She should have to go thru the same approval process that the guy went thru. Hopefully you do a criminal background check as well as credit? How often does the mother not get custody of the kids? Red flag! Probably the criminal background check will explain that.
I had this same scenario happen to me but it was the girl with a young baby on the lease. Every time I drove past the rental early in the morning there was an extra pickup truck in the driveway. Confronted tenant and it was the baby’s daddy. Told her we had to do a criminal background check and credit check on him. He was a felon (sex) with poor credit. The rental has a daycare 250 feet away. He goes or you both go! They were gone in 10 days.
As far as not putting her on the lease goes & then claiming trespassing, all she has to do (my state laws) is get some mail there & she has tenant rights. Easier to evict with her on the lease after Covid.