Yes, whoever provided more than half of the financial support can claim the child. TANF is considered third-party support. Here are the tests of a qualifying relative to see if the person you are referring to can claim your child (from: Who can I claim as my dependent?:(
"Qualifying relative:
- They don't have to be related to you (despite the name).
- They aren't claimed as a dependent by someone else.
- They're a US citizen, resident alien, national, or a Canadian or Mexican resident.
- They aren’t filing a joint return with their spouse.
- They lived with you the entire year (exceptions apply).
- They made less than $5,050 in 2024.
- You provided more than half of their financial support."
Typically, you will be qualified for TANF benefits based on the living situation of the members of the household, but since this is administered by the state I would suggest you confirm the eligibility requirements with your caseworker.