Is a mother required to tell the father if she is living with someone and what other info does she have to share?

The child is 4 years old and lives in New York with mother. It is believed the mother is living with a convicted felon. The father lives in St. Louis, MO and has the child 3 months out of the year. The father pays child support, health insurance and travel expenses but there is no court order for support. The first time in court the judge said to work things out outside of the court room.