Who is the custodial parent? Who do the children live with?
Are you the custodial parent? Do you have an agreement with the other parent to allow the other parent to claim them--due to divorce or that you live apart and share custody? Did one of you sign a Form 8332?
If there is a signed 8332 then the custodial parent retains the right to file as Head of Household, get earned income credit and the childcare credit. The non-custodial parent gets the child tax credit for children under the age of 17.
As far as the IRS is concerned, the custodial parent is the one with whom the child spent the most nights during the tax year--at least 183 nights.
No. If your ex-wife claims the child as a dependent, she is the one entitled to the child tax credit additional or not. It does not matter who the custodial parent is.
Very briefly:
If you live apart and share custody, only the parent where the children live more than half the nights of the year (183 or more nights) is automatically entitled to claim the children as dependents. The IRS does not follow court custody orders, only where the child actually physically lived. In some cases, the custodial parent may allow the non-custodial parent to claim one or more children as dependents. This requires that the custodial parent give the non-custodial parent a signed form 8332 dependent release, that the non-custodial parent must physically mail to the IRS after e-filing.
In this case, only the custodial parent can use the children to qualify for head of household and the child and dependent care credit. However the child tax credit and the additional child tax credit go to whichever parent claims the child as a dependent.