I claim my child has a credit for other dependent but not child tax credit

The only parent who has the right to claim a child as a dependent is the parent that the dependent spent the most nights with.  That parent (the custodial parent) is entitled to the Child Tax Credit, and also to file Head of Household, claim the Earned Income Credit, and the Dependent Care Credit.  The other parent (the non-custodial parent) cannot claim the child.  

If the custodial parent is supposed to give the dependent exemption to the non-custodial parent in certain years, that is done by the custodial parent giving the non-custodial parent a signed form 8332 Dependent Release.  This allows the non-custodial parent to claim the Child Tax Credit and exemption only.  The ability to qualify for Earned Income Credit, Head of Household, and the Dependent Care Credit, always stays with the custodial parent and can't be waived, shared, or transferred.