for federal income tax purposes, in order to claim Head of Household, you would need to have a dependent that is biologically related to you. It could include most all blood relatives, except cousins. In IRS terminology, it's a "Qualified Person".
A domestic partner would not qualify you for HOH filing status as a domestic partner is not a Qualifying Person. This is the federal rule.
However, if you provided more than 50% of their support, they lived with you ALL YEAR, and they earned less than ~$5200 (2025), you could claim this indiviudal as a dependent which could create up to a $500 Other Dependent Tax Credit.
State rules may vary.
p.s. what you do on a W04, does not ncessarily matter. What matters is what you do on the tax return and do not owe that much money comes tax time.
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