He doesn’t have a personal private plane. The Pope typically flies on chartered commercial aircraft arranged by the Vatican, most often with Italy’s ITA Airways, and that aircraft is sometimes nicknamed “Shepherd One.”

What that means

  • It is not a plane he owns.
  • The Vatican charters the aircraft for each trip.
  • For some journeys, the return flight can use the host country’s national airline instead.

About Mark Pope

If you meant Mark Pope the Kentucky basketball coach, the public post I found refers to “the private plane Mark Pope and the Kentucky basketball staff use for recruiting,” which suggests a chartered team/recruiting flight rather than a personally owned plane.

Quick answer

So, the clean answer is: neither the Pope nor Mark Pope appears to “use” a privately owned plane in the usual personal-ownership sense; both are described in public sources as using chartered flights for official/team travel.