Air carrier Captain reported a course deviation occurred during departure from JFK as a result of the climb fixes not being properly loaded in the FMS. ATC provided vectors to a fix and handoff to another controller.

Narrative

After departing 31L and switching over to departure ATC gave us a heading. After we made the turn ATC asked which departure we were on. I replied that we were on the Kennedy 5; we were given another heading and asked again which departure we were given. I again restated that we were on the Kennedy 5 departure. They replied saying we were supposed to be on the Breezy Point [Climb] then gave us direst to a fix and switched us to another controller.When inserting the departure in the FMS and when the JFK5 departure off of 31L was selected no transition was prompted. I missed that the Breezy Point Climb was on the PDC when reviewing the route. Once airborne the FO (First Officer) flew the loaded right which didn't include the Breezy Point Climb fixes. This was my first time flying out of JFK in years and missed this part of the departure while reviewing all airport information.Having the different climbs preloaded or no part of the departure being in the FMS would have help trap the error. Having Tower give the climb name with takeoff clearance would have help catch this as well. A better review of the PDC by me would also help prevent and catch this error.