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.
This answer is provided under the Avvo.com “Terms and Conditions of Use” (“ToU”), particularly ¶9 which states that any information provided is not intended as legal advice or to create an attorney-client relationship between you and me or any other attorney. Such information is intended for general informational purposes only and should be used only as a starting point for addressing your legal issues.
In particular, my answers and those of others are not a substitute for an in-person or telephone consultation with an attorney licensed to practice in your jurisdiction about your specific legal issue, and you should not rely solely upon Legal Information you obtain from this website or other resources which may be linked to an answer for informational purposes. You understand that questions and answers or other postings to the Site are not confidential and are not subject to attorney-client privilege. The full Avvo ToU are set forth at http://www.avvo.com/support/terms .
In addition, while similar legal principles often apply in many states, I am only licensed to practice in the State of New York and Federal Courts. Any general information I provide about non-New York laws should be checked with an attorney licensed to practice in your State. Lastly, New York State Court rules (22 NYCRR Part 1200, Rule 7.1) also require me to inform you that my answers and attorney profile posted on the Avvo.com site may be considered "attorney advertising" and that "prior results do not guarantee a similar outcome".
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.
* If you found my answer to be helpful, or the "best answer," please feel free to mark it accordingly.
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.