Gifts received are not income and more particularly they are not "earned income", for the purposes of claiming the refundable child based tax credits.
Since you will receive no benefit from claiming your children, on a tax return, the next question might be can your father claim you and the children. Probably yes. But, he can only claim the more generous credits if you and the children lived with him for more than half the year.
There are two types of dependents, "Qualifying Children"(QC) and Other ("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 and the Child Tax Credit. They are interrelated but the rules are different for each.
The support test is different for each type. The support test, for a QC, is only that the child didn't provide more than half his own support. The support test for a Qualifying Relative is that the taxpayer provided more than half the relative's support.
See full dependent rules at: https://turbotax.intuit.com/tax-tools/tax-tips/Family/Rules-for-Claiming-a-Dependent-on-Your-Tax-Ret...
