Yes, there are restrictions. Learner permit holders aren't allowed to drive on any street within a park in NYC or on any bridge or tunnel under the jurisdiction of the Tri-borough Bridge and Tunnel Authority.
Can you operate a moped with a motorcycle learner permit?
Yes, but the same motorcycle passenger rules apply to mopeds, as well, meaning the only passenger allowed to ride with you is your supervising driver.
What about a motorcycle learner permit Class M license?