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.
If you are living together unmarried, you must delete the child completely from your return. When entering a dependent, the question "do you have a custody order" applies only to a court order for parents who are divorced or live apart and share custody, not for informal agreements between unmarried parents to live together. The parent claiming the child must answer "no" to the custody agreement question to get the full benefits of the dependent, and the other parent must delete the child completely.
If you are divorced or separated and sharing custody, and you have custody more than half the nights of the year, then it is proper that the child still qualifies you for EIC and head of household status. The only tax benefit that is transferred to the other parent with form 8332 is the child tax credit. The ability to qualify for EIC and HOH always stays with the parent who has custody more than half the nights of the year.