It's a little hard to answer your question because you seem to be mixing up two different terms -- "Medigap" and "Part C" -- so I'm not exactly sure what Medicare Part C plans, called Medicare Advantage, are insurance plans, usually a kind of managed care, that some people enroll in to replace traditional Medicare Part A (hospital or inpatient coverage) and Part B (medical or outpatient coverage). A Medigap insurance policy, on the other hand, is a private insurance policy that people can buy to provide supplemental coverage when they are enrolled in traditional Medicare Part A and B. If you have a Part C Medicare Advantage plan, you don't need a Medigap policy. It sounds like you're talking about a Part C Medicare Advantage plan, but in general the answer will apply regardless of which one your father has.
insurance you're referring to.
When someone enters a nursing home, the nursing home takes care of daily living and nursing needs, but NOT medical care. For that, your father still needs the services of doctors and, if needed, a hospital. And these doctors and hospital need to get paid separately, in addition to, payment to the nursing home. So, a resident in a nursing home still needs medical insurance, whether Medicare or otherwise.
In your father's situation, he's qualified for Medicaid coverage of his residence in the nursing home. Often, that means he's also eligible for Medicaid medical coverage. But the two Medicaid programs -- nursing home coverage and medical coverage -- have slightly different eligibility rules. So, you need to find out from his nursing home whether he has also qualified for Medicaid medical coverage. If not, you need to help him apply by contacting the Michigan Medicaid agency. If he qualifies for Medicaid medical coverage, Medicaid will pay for most of his medical needs, including the monthly premiums for either traditional Medicare Part A and B, or for certain Part C Medicare Advantage plans. (If he has a Mdigap policy, however, he would not need to keep it -- Medicaid medical coverage would replace the Medigap policy by covering the costs that Medicare doesn't pay.) But if he qualifies for Medicaid medical coverage, you then have to find out whether his current Part C Medicare Advantage plan -- the one you're now paying for -- is covered by the Michigan Medicaid program. If so, then they should pay the premium, not you. If not, your father will have to decide whether to switch back to traditional Medicare Part A and Part B (with Medicaid paying the premiums) or to change to a different Part C Medicare Advantage plan that Michigan Medicaid does cover.
If for some reason your father does not qualify for Medicaid medical coverage -- and until he is enrolled in Medicaid if he does qualify -- then he needs to keep his Part C Medicare Advantage plan (or some other Part C plan, or return to traditional Medicare Part A and B) in order to have health insurance. If so, you or he will have to continue paying the premium.