It is not HIS money; you are a family, a team, and both of you work. Right now it has apparently been a mutual decision that he's the one to bring in money and you're the one to handle most/all the domestic labor.
You are in an abusive, controlling, manipulative relationship, and the fact that he's convinced you it's "his" money is only the tip of it. When you object, he knows how to "punish" you to keep you quiet. This is a terrible situation for you, and a damaging thing for children to experience.
IF you can get into marriage counseling, that would be a great place to start, and I would say there's still some hope for the growth and maturity you need from him. But I'm assuming counseling is financially out of the question, and that your husband wouldn't agree to it anyway. If you can't get counseling, then you will never have it easier than where things are now, and that they will probably only get worse.
Having been in a somewhat less controlling but still abusive marriage myself, I would strongly advise you to leave, the sooner the better. If you have family you can go to, take your children and go on a day when he's away. If there's no friend or family to take you in, check out women's shelters in your area. Call a cab or a friend for transportation.
Do, do it, DO IT! The sooner, the better. I am not overreacting here. I've seen too many women harmed and too many children traumatized by this kind of relationship. Don't give him advance warning, don't tell him where you've gone, and don't, under any circumstances, go back to him until he can prove he's in counseling and making workable changes. (That will probably be never.)
If you make a decisive enough move, your husband might be startled into seeing how unfair your marriage has been. While you are apart (and don't fool yourself into thinking a short separation, an apology, and all will be better), get counseling for yourself, too. I'm guessing that you struggle with low self-esteem, or you wouldn't have let the situation get this far, you wouldn't think in terms of having to beg for his money, and you would simply expect a more equal partnership.
Good luck. My heart aches for the difficult choices you face. The good news is, you can learn from this, become more whole, have a more satisfying life, and have a chance at a better mate in the future.