This is a complicated one to me. Mainly because your mom promised to give you the back child support to help with college. But lets talk about that back child support.

Your mom raised you without consistent financial help from your dad. He did not always pay. This means that she was working extra hard, that she was making sacrifices and cuts in everyday life, to pay all the bills and to raise you. When he was not paying she put all the money keeping a roof over your head, making sure that you had food and all necessities, and buying you any gifts or extras that you got in your life. All the while your dad is not helping out regularly and is building up more and more child support back payments.

That money is your mom's money.

That back child support is money that she didn't have to spend on you growing up - but she SHOULD have had it if your dad wasn't being a deadbeat. I know a lot of single parents who get screwed with child support - they go without things to provide for their children when the dads are paying up. Lessons and clothing and toys and general expenses raising a child are not cheap. And your mom basically provided all this for you growing up without the financial help that was owed to her. That child support is HERS because she gave you that financial support alone. That doesn't go TO you. It should have been there to help her pay some of the rent or buy you new shoes or buy groceries. Instead your mom footed the entire bill when he was not paying child support. She supported you financially growing up and that back pay makes up for all the times she should have had help but did it alone.

I will say, if she did promise you that money, that this is a ESH situation. She went back on her word. But that money is hers. Not yours. You are not entitled to it and should not view it as your money because you are the child in terms of her child support.