If you claimed the child on your 2020 tax return as a dependent then they would be eligible for the Advance Child Tax Credit.
You should be able to file as Head of Household in 2021 if you are not married or have not lived with your spouse at any time during the last six months of the year and claiming the child as a dependent.
See this TurboTax support FAQ for Head of Household - https://ttlc.intuit.com/community/family/help/do-i-qualify-for-head-of-household/01/25539
