So it is a public transport flight that can be operated by a PPL
No, According to Article 159 its a PRIVATE FLIGHT by Exception!
Public transport and aerial work - exceptions - charity flights
159 (1) Subject to paragraph (2), a flight shall be deemed to be a private flight if the only valuable consideration given or promised in respect of the flight or the purpose of the flight other than:
(a) valuable consideration specified in article 157(3)(c); or
(b) in the case of an aircraft owned in accordance with article 162(2), valuable consideration which falls within article 162(3); or is given or promised to a registered charity which is not the operator of the aircraft
and the flight is made with the permission of the CAA and in accordance with any conditions therein specified.
(2) If valuable consideration specified in article 157(3)(c) is given or promised the flight shall for the purposes of Part 3 of this Order (other than articles 19(2) and 20(2)) be deemed to be for the purpose of public transport.