Can I file as head of household if I support my son's father who also lives in my home ? , we were never married and he does not work

Your BF is not a qualifying dependent for head of household.   If your son is your dependent--your own son may be your qualifying dependent to file as HOH.

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 filing status (single or married filing separately) into MyInfo, and then enter your qualifying dependent, TurboTax will offer HOH as your filing status.

If your BF lived with you for all of 2020 and had less than $4300 of income (not counting Social Security) then you might be able to claim him for the $500 credit for other dependents.   You do not get stimulus money for claiming him.

CREDIT FOR OTHER DEPENDENTS

https://ttlc.intuit.com/questions/4499708-what-is-the-500-credit-for-other-dependents-family-tax-credit