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The court, when considering requests for modification of child custody/visitation, will look to the best interests of the child. So, in the case of an absentee parent, the court will look to see what the parent's motivations are in all of a sudden requesting visitation. Because visitation with essentially a stranger can be uncomfortable and frightening for the child, the court may even order supervised visitation: https://www.childsupport.ny.gov/dcse/access_visitation.html
Also, the court, if they do grant visitation, will likely do so in small increments - a few hours at a time- so that the child may become use to the absent parent. This assumes that the parent is a fit and suitable parent. If the relationship progresses, the court will gradually increase visitation time. If the child is older (generally 12 +) the courts will often ask the child's opinion on the matter and take that into consideration.
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