I'm jumping on the don't-give-him-any-money bandwagon. Prayers are free and you can give him plenty of those, but don't COUNT on him for anything or GIVE him anything. I know you are praying for a reconciliation but in the meantime you need to file for a legal separation and get things on paper for yourself as well as your kids. That he even ASKED you for money I find repugnant. You and your children can find a better man than that. How dare he? Stand up for yourself honey, don't let him treat you like that. Praying for him is one thing but you ain't gonna get him back letting him stomp all over you. He was obviously embarrassed to ask his new lady for money and you just saved his butt from that, kept him looking good. Don't do that again.
You need a legal arrangement about how much money he is going to give you exactly when each month for the support of your children and to help you while you establish yourself in a new residence. That was HIS request after all. That his family is supportive of you and not him is really immaterial from a legal standpoint, he might not like it, but it is good for the stability of your children not to lose his FAMILY while he has his girlfriend. We are all trying to tell you that legally you hold all the cards and you are not playing them.
I understand that you do not want to divorce him, but you need a legal separation. You could still be responsible for his debts! You want to pay for his girlfriends night out on the town? Help with her rent? Does he have any credit cards with your name on them? Where is all that money going? Seems like his kids deserve some of it. How can you provide for them if he takes his good salary and can't even budget his way through while you manage.
I know you are praying that he will change his mind, but he doesn't sound like a "keeper" to me. Personally I would pray for a better role model (yourself) to be in charge of your children without the interference of this person. There will be other male role models in their life that will be responsible role models for them for them to compare him to, and behavior to leave all of you for another woman is really way past the time to be hoping for a reconciliation, and way beyond what you should tolerate.

Just my opinion, but I'm trying to help you not see you hurt further.
People only give you what you demand of them. You need to have him be a man. If he wants his cake and to eat it too, well, he is gonna be one broke guy. Such is life for someone who leaves their family. Let reality slam him in the face. I'd tell him not to let the screen door hit him personally. He sounds less and less of a good guy the more you say about him. I'm sorry hon, but I don't think God means for you to endure this to show your faith in him. I think He is showing you that you need to move on, this is not the man to raise your children right, as much as you want him to be. You can't be the pillar holding up a marriage and if he has another woman, he has essentially told you the marriage is over already. Men never like clingy anyway, so it won't work sweetheart, you make yourself less attractive that way, not more attractive (it is the perverse male mind to want what they can't have). I wish having faith in him and praying for him would turn every bad man good, but that just isn't how it is or else the world would be a lot better place. Sometimes we have to move on to find the good in our lives, spend it with the good people. Life is short, don't waste it, or worse, teach your children that this behavior is forgivable. You have done nothing wrong, and don't deserve this, and you have given him every chance. I'd think about that long and hard, and while I'm doing it, file those separation papers.
