2 children, parents live together, but aren't married. Who can claim, and whose health insurance?

Our current situation is we have 2 children. 
My partner, their mother, is self employed, and has a lot of write offs, so her income looks very low. 
Because of this, she qualifies for our state Medicaid for health insurance, and thus has both of our children on it as well. She has been claiming our 1st child (3yrs old), but now that we have 2  (2nd born in Aug. '24) we're questioning if I should be claiming one of them or both and as head of household?

Can I claim my kids on taxes, with them on their mother's Medicaid health insurance?