Are you living together unmarried, or living apart and sharing custody?
When living together unmarried, the parent who is not claiming the child must delete the child completely. There is a custody question which, because of the way it is worded, can be very confusing. The custody question only applies to a legal or court order between parents who live apart and share custody. It sounds like you indicated the child lived with you and that you would let the other parent claim the child with a custody agreement. That caused you to be eligible for HOH, but the other parent now can't file. This kind of "splitting" the dependent is only allowed for parents who live apart and share custody.
If your plan is for the other parent to claim the child, they have to answer that the child lived with them, and there is no custody agreement. They will be blocked from e-filing, so they need to file by mail. After your refund is paid, you will need to amend your return to delete the child completely. You will have to pay back part of your refund. No matter when you amend, the e-file block on the other parent won't be cleared this season so they have to mail their return no matter when you amend.
Also note, HOH requires that the parent both claim the child AND pay more than half the household expenses. Exactly half is not enough. If you truly paid more than half the expenses, then the other parent paid less than half, and can't file as HOH even with the child, they still file as single.