I have been with my boyfriend for 11 years...
Is there a reason you didn't get married at some time over the last 11 yrs? Especially after having 2 children, many (though not all) choose to get married.
Honesty and faithfulness are very important qualities to most who are committed couples. If you no longer want to be with him, they wouldn't necessarily grow up without
a father. You could kick him to the curb if you see no future as a couple, but not cut him out of their lives.
BTW, generally speaking, good fathers do not lie and cheat on their partners.
You could ask him to go to couple's counseling, as a possibility. However, if you no longer want him in your home as your partner, he's still their father but he could visit them but not live with you. There ARE alternatives.