There is much you don't say.
(1)
If you are 19 or older and not a full time student, or age 24 or older and not disabled, your mother can't claim you as a dependent unless she paid more than half your living expenses and your taxable income is less than $5050. It doesn't matter where you lived. If you are between 19 and 23 and a full time student, your mother can claim you if you lived in her home more than half the year and you did not provide more than half your own financial support.
(2)
If you can be claimed, you must check the box that says "I can be claimed by another taxpayer", even if you don't want to be claimed, and you can't claim dependents of your own.
(3)
If your children lived in her home more than half the year, and you can be claimed as her dependent, then she can also claim the children who lived in her home. If your children lived in her home more than half the year, and you can't be claimed as her dependent, then either one of you can claim the children. You can divide them any way you like (all or nothing, or shared) subject to these rules:
a. If you want to claim them, you can. The parent gets absolute priority.
b. Your mother can't claim them even if you are willing, unless her taxable income is more than yours.
(4)
If you have 4 children, where do they all live? The biological father can't claim them as dependents at all, unless, either
a. They lived in his home more than half the year, or
b. You give him a signed dependent release form 8332. With the form, he can claim the $2000 child tax credit, but he can't file as head of household or get EITC, because those benefits always stay with the parent where the children live, and can't be transferred or shared.
(5)
If you want to decide how to claim your children (between you and your mother) to get the best tax advantage, you will have to test different scenarios. There is no right answer. It depends on your income, whether you are eligible for EITC, and other factors.
