This may be possible depending on whether it can be accomplished with minimum disruption to your child's other activities, particularly school and any extracurricular activities she's engaged in.
You don't say anything about the age of the child or your current custody or visitation arrangements or court orders, or what your two days a week off are (weekdays, consecutive, are they always the same two days off?, etc., where the mother lives and whether you live in the same general area) but it may be possible to successfully handle this visitation. A lot depends on the particular circumstances and how practical the arrangement you are proposing would be. It would go easier with the mother's consent.
You would accomplish this result by petitioning the family court for modification of visitation. The family court wants to provide non-custodial parents with a reasonable amount of visitation so each parent can develop as normal a relationship with a child as possible, but the focus is on the child's needs and the practicality of the arrangement. There are no "fathers rights" per se. If your proposal is disruptive of the child's routine in a way that doesn't work well for the child or the mother, it will be opposed in the court proceedings and you will find yourself with few rights other than the minimalistic several evenings a week and alternate weekends, etc.
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Generally a non-custodial parent gets alternate weekends, a split of holidays and a dinner visit mid-week. But while that's what's generally done, there are numerous exceptions to that general rule. As such, to determine whether what you want would translate into reality, one would need to do a full assesment of your situation. Thus, I highly suggest you scheule a follow-up consultation with a NYC Child Custody lawyer.
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You would have to submit a petition for visitation in family court, or modification of visitation if you already have a visitation order from the court.
Remember, as a father you have absolute rights however the courts main question will always be the best interest of the child so the court will not make decisions that would cause harm to the child in any way... Ie pulling the child out if activities or classes.
Much luck sir
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