ITA has round-trip flights from New York (JFK) to Palermo (PMO) for $382 with a connection. Flights October to March. Departures on Monday, Tuesday, Wednesday, and Sunday. 7-day minimum stay required.
Some small OTAs on Skyscanner have this for as low as $328; which is cheaper than the airline. If you plan on booking there, beware of any site that charges a fee to cancel a ticket within the 24-hour “free” period. Use unknown booking sites at your own risk.