There are a lot of possibilities and you haven't given enough facts to really answer the question.
Is this a child you share (you are the two biological parents)?
Are you talking about Affordable Care Act subsidies (Marketplace insurance)?
Do you live together unmarried and share custody? Certain rules that apply to divorced or separated parents may not apply when you live together.
If your partner has medical insurance through work, they can cover their biological child even if they don't claim the child as a tax dependent. However, if the child is not their biological child and not their tax dependent, there can be significant tax consequences for the employer and the employee.
We really need more information.
