You can claim Head of Household (HoH) filing status, regardless of whether your girlfriend and/or her child qualify as your dependent. You can claim HoH because YOUR child is your qualifying person. Your child is your qualifying person even if she is not your dependent, because you are letting the other (non custodial) parent claim the child.
Yes, HoH is very beneficial (lower tax rates and higher standard deduction).
But, your girlfriend and/or her child almost certainly do qualify as your dependents because "girlfriend has no income". .The child may qualify as your dependent, but because he is not related, he cannot be a qualifying child for
the earned income credit, child tax credit. But, you do get the $4050 exemption (deduction from income) for each, before tax is calculated
