Are the children YOUR own bio-children? If so, yes, you can claim the children as your qualifying dependents and file as HOH. If your GF has less than $5050 of income (not counting any Social Security) and she lived with you all year, you may also be able to claim her and get the $500 credit for other dependents for claiming her as a qualifying relative.
Am I Head of Household?
https://ttlc.intuit.com/questions/1894553-do-i-qualify-for-head-of-household
https://ttlc.intuit.com/questions/2900097-what-is-a-qualifying-person-for-head-of-household
If you qualify as Head of Household, when you enter your marital status (single or married filing separately) into MyInfo, and then enter your qualifying dependent, TurboTax will offer HOH as your filing status.
CREDIT FOR OTHER DEPENDENTS
https://ttlc.intuit.com/questions/4499708-what-is-the-500-credit-for-other-dependents-family-tax-cre...
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