To have Head of Household filing status you must have a qualifying person as your dependent. Neither your girlfriend or her child will qualify you for HOH since they are not related to you. Your child would qualify you for Head of Household.
You may be able to claim both of them as your dependents under the Qualifying Relative rules even though they are not related to you. They must meet all the requirements under the rule.
To be a Qualifying Relative -
1. The person cannot be your qualifying child or the qualifying child of any other taxpayer. A child is not the qualifying child of any other taxpayer if the child's parent (or any other person for whom the child is defined as a qualifying child) is not required to file an income tax return or files an income tax return only to get a refund on income tax withheld.
2. The person either (a) must be related to you or (b) must live with you all year as a member of your household.
3. The person's gross income for the year must be less than $4,050 (social security does not count) in 2017
4. You must provide more than half of the person's total support for the year.
5. The person must be a U.S. citizen or a U.S., Canada, or Mexico resident for some part of the year.
6. The person must not file a joint return with their spouse.
