@JJOTOOLE Since the grandchild did not live with you for more than half the year, you can not get the $2000 child tax credit. You would only be allowed the (up to) $500 Other dependent credit.
There are two types of dependents, "Qualifying Children"(QC) and standard ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, student status, a relationship test and residence test. Only a QC qualifies a taxpayer for the Earned Income Credit (EIC) and the Child Tax Credit. They are interrelated but the rules are different for each.
So, you also can't claim the child tax credit (CTC), on the other child, that doesn't live with you.
See the qualifying relative rules above. One clarification. It says: They aren’t a dependent on someone else’s taxes. It should say they are not the QC of another taxpayer. So, if the mother is required to file a tax return, you cannot claim her kids.
I assume the father also does not live with the child. There are special rules for divorced and separated parents, so he can claim the CTC (but not EIC). But, he can not do so without the written permission of the mother, on IRS form 8332. So, the father has no chance of getting a court or the IRS to award him the tax benefits (my non lawyer opinion).
I assume the mother does not need to file a tax return. So, she should probably let the father claim the child for $2000, rather than you for only $500. She can probably get the IRS to offset it for back child support, by going thru the local child support agency.
