Can I keep my spouse from exposing our children to his new girlfriend (her new boyfriend)?

Generally, no, unless there is something about that person that makes association with them not in the best interest of the children. Each party is entitled to have the society of other individuals during their time of sharing.

Politically, in the very early stages of the divorce, it is always smart to realize that involving the children with a new boyfriend or girlfriend will cause resentment and hostility in the former spouse.

This hostility will then be converted into acrimony and perhaps an enormous amount of needless expense associated with attorney's fees. It is therefore best to use caution in this area at the beginning stages of the divorce.