No, because you need a dependent to claim HOH.
You would need to list one or both children on your return as dependents, and file by mail since e-filing will be blocked. Your GF will have to file an amended return to remove the children and pay back part or all of her refund.
In the dependent and custody issue, answer "No" to the question about whether you have an agreement with the other parent about who will claim the child. This only refers to legal custody agreements between divorced or separated parents, and answering yes will send you to the wrong kind of interview questions for your situation.