Can another person claim a child on ssi?

Maybe.  There are 3 things you need to know for this situation.

1. Even if you had no taxable income this year, you would qualify for a $3000 child tax credit (or $3600 if the child is age 5 or under) if you filed and claimed your child as a dependent.  Also a $1400 stimulus payment (EIP round 3) if you did not already receive it.  For 2021 only, the full child tax credit can be refunded to you even if you did not work.

2. If the child lives with you more than half the nights of the year, the only way that the other parent can claim the child is if you give them a signed form 8332 dependent release.  This allows the other parent to claim the child tax credit, and maybe a stimulus payment (for 2021 only), but not the dependent care credit, head of household, or earned income credit.  Those benefits only go to the parent who has custody more than half the year, and those benefits can't be waived, transferred or shared.

3. SSI for children is based on both disability and financial need.  The government looks at the financial resources available to the child.  If the other parent already pays court-ordered child support, then that should have already been factored into the eligibility test, so if the father claims the child as a dependent, that should probably not affect the child's eligibility for SSI.  However, if the father is not paying child support, the fact that the father claims the child could cause the state to take a new look at your situation.  

(In the case that you live together unmarried with the father, the father can claim the child as a dependent without a form 8332 and the father will also qualify for EIC or head of household status.  However, this will definitely cause the state to review you child's case, since if you and the father are living together, the father's income is supposed to be declared when calculating the child's financial need.)

The father can claim the child if you give him a signed Form 8332. Since the child lives with you I don’t think that the SSI will be affected but I would defer to someone more knowledgeable about that. 

You should know that you can receive child tax credit if you file and claim the child even though you have no income. This is a change in the law for 2021. You would get $3,000 for a child between ages 6-17 and $3,600 for a child under age 6.