If you are not legally married to her in 2020 then only one of you can claim the child on their tax return.
To be Head of Household you must provide over one-half the cost to maintain the home and have a qualified person as a dependent - https://ttlc.intuit.com/community/family/help/do-i-qualify-for-head-of-household/01/25539
The child must be living in your home in 2020 so if your current place of residence is not in the home of the mother of the child you cannot claim the child on your tax return.
To be a Qualifying Child -
1. The child must be your son, daughter, stepchild, foster child, brother, sister, half brother, half sister, stepbrother, stepsister, or a descendant of any of them.
2. The child must be (a) under age 19 at the end of the year, (b) under age 24 at the end of the year and a full-time student or (c) any age and permanently and totally disabled.
3. The child must have lived with you for more than half of the year. Temporary absences while away at college are considered living with you.
4. The child must not have provided more than half of his or her own support for the year.
5. If the child meets the rules to be a qualifying child of more than one person, you must be the person entitled to claim the child as a qualifying child.
6. The child must be a U.S. citizen or U.S., Canada or Mexico resident for some portion of the year.
7. The child must be younger than you unless disabled.
