Medicaid Investigation

Has anyone experienced Medicaid investigation? The father of my unborn child is an on again off again ordeal. We’re currently together, but I didn’t list him when applying because we weren’t at the time. I’m due in July, and hesitant to put my infant on Medicaid if I could receive jail time. If I knew for sure the father would support us and be with us long term I wouldn’t hesitate to list him, but he is paid in cash and isn’t exactly here all the time. So even if I do list him he could be sought by the IRS of I would be left broke supporting myself and the baby. My parents are terrified I’m going to be sent to jail if I don’t include him, but put him on the birth certificate or give the baby his last name.