If the adult child has gross income of $4,700 or more in 2023 then you cannot claim them as a dependent and would not be eligible for Head of Household filing status.
If you are referring to tax year 2022 the gross income test is less than $4,400 to claim an adult child as a dependent under the Qualifying Relative rules.
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 $4,400 (social security does not count) in 2022
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.
See this IRS website for calculating support - https://apps.irs.gov/app/vita/content/globalmedia/teacher/worksheet_for_determining_support_4012.pdf
