In NY, anything that get stamped on as a restriction on a pistol permit is considered a bureaucratic limitation. The NYS Court of Appeals ruled a decade ago that while there's nothing in the law allowing issuing authorities to do this and there is no law to back up any "violation,"
it can be used as a reason by the issuant to revoke it, as NY is a may-issue state.
So, if your permit is marked "TARGET SHOOTING AND HUNTING ONLY" (as is that of my brother in Albany), you're free to carry anywhere not prohibited by law... but if you get stopped and you have to present your permit, you might have it revoked as a violation of a bureaucratic restriction. And beyond that, it might be a problem for you if a cop has no clue about NYS carry laws, and not many do; they might think it's a law and arrest you until they have it sorted out. If they do, there are no laws to charge you with, so you'll be let go after plenty of time has been wasted. Just one of the many
reasons NY laws need to be changed.
(Edit) Addendum - Forgot to tell you... many issuing judges will issue all new permits with target/hunting sort restrictions as a template, but will then remove them officially if you complete some sort of advanced handgun training. In the Albany area, where getting a permit in the first place can be a troublesome experience, they'll do so if you complete any of the "advanced" level courses offered by the NRA or by some local police stations. I'd contact your county judge and see if he/she would accept that as a means of doing so.