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If the mother works nights, there is a special rule to count the days as if they were nights.

Only the custodial parent by the IRS rules in the link above can claim the dependent care credit, EIC or Head of Household.  The non-custodial parent can only claim the exemption (dependent) and child tax credit.

For a qualifying child, who pays the support is not a consideration whatsoever, as long as the child does not support him/herself.
**Disclaimer: This post is for discussion purposes only and is NOT tax advice. The author takes no responsibility for the accuracy of any information in this post.**