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It would be considered Medicaid fraud to under-report your income in order to obtain medicaid. Therefore, it could be considered Medicaid fraud if your son's access to medicaid funds is based on the salary that you are reporting
You should hire a local attorney to help you correct this under-reporting and have them explain, if the issue comes up, that it was an oversight.
Paying the nurse on the side to babysit your other children would not be medicaid fraud as long as she is not billing to Medicaid any services that she provides to your other children.
And you can have both private insurance and Medicaid. If you have private health insurance available to you or a member of your family, eligibility for Medicaid will not be affected. Any service not covered or only partially covered by a private health insurance plan and listed as a Medicaid covered service will be paid by Medicaid. Medicaid will pay up to the maximum Medicaid is allowed to pay
Whenever you have both private health insurance and Medicaid, you should show both your Medicaid card and your health insurance card to your medical provider each time you receive services.