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Assuming that the mother lived with the child the majority of the year then:

 

If you are the non-custodial parent where the child did not physically live for more than half the year (183 nights) then:

When you enter the dependent, you say that he is "Your child" (not you and new spouse if remarried),
He/she lived with you 0-5 months (less then 6 months),
“no” the child did not pay more than half of his/her own support,
"yes", you have a custody agreement, and
"yes” I am claiming this year.

That will give you the child as a dependent, and child tax credit.  
 
The EIC, and Child Care Credits can only be claimed by the parent where the child actually lived.    

**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.**