A child can only be claimed as a dependent by one individual. However, there are special circumstances for children of divorced parents.
Because we are in an age of technology, the children's' social security numbers are flagged to let the taxing agencies know that they were already claimed by someone else.
If your children are not your dependents for the current tax year, and you haven't filed your return, you should remove their names from your tax return. However, if the return was filed, you should prepare an amended return (Form 1040X) to exclude them as your dependents and make the necessary adjustments.
Your mother should also file an amended return if she can claim the children as dependents, and she meets the qualifications to claim the child tax credit.
I have attached two Internal Revenue Service interactive links that will help you determine if you or your mother are entitled to claim the child tax credit, and/or your children as dependents.
IRS
ITA
Claiming
Dependents
