He also pays the dependents health insurance

HOH can only be claimed by the person with whom the child lives with for over half the year.

If he only has the child every other weekend, the best he could get would be to claim the child as a dependent and get the CTC, but the custodial parent (the one the child lives with more than half the year) would need to sign an 8332 releasing the child's dependency to him. 

The custodial parent would still qualify for HOH and and child care credit for that child, only CTC can be released to he non-custodial parent.