Is marriage to the aforementioned girl considered forbidden, and what must the questioner do, and is there any repentance for him after he fathered children with her, knowing that he lied to his mother with a vision in order to marry her, motivated by wanting to bring her out from the people of innovation to the people of Sunnah?

Lying about a vision has already been forbidden. The door to repentance is open, so one must hasten to it, conceal oneself, avoid informing the mother of the deed, and increase good deeds, as Allah (the Most High) says: "{Your Lord has decreed upon Himself mercy: that any of you who does wrong ignorantly, and then repents after that and corrects himself - then indeed, He is Forgiving and Merciful}" [Al-An'am: 54], and His saying: "{But whoever repents after his wrongdoing and reforms, indeed, Allah will turn to him in forgiveness. Indeed, Allah is Forgiving and Merciful}" [Al-Ma'idah: 39]. This lie does not affect the marriage contract.