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The $4,050 tax break for a child that you see is not a refund.   It is the child's exemption that reduces your taxable income by that amount, but not less than zero.  If your taxable income is already zero then it cannot lower it further and would not change your refund.
**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.**