First of all, learn a few basic facts of law: (1) Just living with somebody and being physically intimate DOES NOT mean it is Marriage. At best, it can be called COHABITATION. (2) Having a child with someone also DOESNOT constitute that it was marriage.
A marriage is leagl only if it was solemnised as customs and traditions acceptable to the society or it was legally registered as per prevailing laws.
As for yuor UPSC , it should not have any effect what you do in your personal life. Its not UPSC's business.
Your status with XYZ can not be called MARRIED. Because the relationship IS NOT legal, there is no question of breaking it legally. It was an emotional relationship, so you need to break up it emotionally ( No need to break up legally)
As for AAdhar card etc. in his name (fake name), he broke the law and you are not responsible unless you were an accomplish in this unlawful making of documents.
Of Course, you can give your child your name, you are the mother...you are the most important relationship of your child.