You are in a very difficult position. No one wants to disrespect their mother, but how can you possibly raise your children in that environment? IF your mother were a stranger, would you allow this to continue?You didn't allow that with the children's mother, why is it ok for your own mother to do the same? I think it is time for you to stand up to her - as difficult as it may be.
I do understand your situation. Men - regardless of their race- have no rights in the justice system or social services in NY. The fact that you ( and other men) are taking responsiblity for your children should be given preferential treatment by the courts, not tossed aside, like the garbage their mothers are.
Take back the kids social security cards. You will need them.
Who's name is the lease/rental agreement in? If it is in your name, then you can have your mother evicted/removed by the courts if necessary. If it is in her name, you have a tougher battle.You will need to find another place for you and your children to live. Try HUD - get on the list for subsidized housing vouchers and see if they can assist with public housing. I know it is not the best option, but it is better than your current situation.
Then go to modest needs , catholic charities, salvation army or 211 for security & 1st month's rent - once you are approved.
Also check with 211 for food pantries and other necessities while you are waiting. There are organizations locally that can assist with your basic needs.
Once you are in your own apartment, you can re-apply for social services. including day care assistance and WIC - you should quailify for WIC because your daughter is 5 and your son is 1.Also apply for HEAP, TANF, food stamps and medicaid.
Also apply for child support - I do not know how that will work with the mom being out of state, but try anyway. You can get a free attorney - they are not the best, but they are free - to help - with any legal matters - if necessary.
If the children's mother is on social security disability - the children may qualify as well.
Also get your GED - you can do this online or in a local highschool - regardless of what you have accomplished this far, you will need it to obtain a good job to raise your children.
Social services or unemployment can help with job training and certification for computers.
I would also suggest Al anon or nar anon to learn how to beat deal with your mom.
You are too young to have this wieght on your shoulders, but you are doing very well. You should be proud of everything you have accomplished. By simply taking care of your children, you stand far above many who turned and walked away. It will be tough, but not impossible. You can do this! You have every right to be proud of yourself. In time, you will loook back on this as just a bad dream.