Why does the Pratt & Whitney 747 have an extra engine on the right side but not left side?
It's an engine testbed aircraft. In this particular picture, the tested engine is probably one of the PW1X00G series (mainly used by Airbus A220, Airbus A320neo, Embraer E-Jets E2, Irkut MC-21, Mitsubishi SpaceJet) and the aircraft is one of two Boeing 747SP testbed aircraft owned by PW.

Engine testbed aircraft test the engines in-flight, with parameters and conditions that can not be tested in a static test on the ground (for example, low temperatures and non-static thrust).

Engine testbed aircraft often end up in unusual configurations. Other examples include (but not limited to):


General Electrics Boeing 747 testbeds: 1 2
Honeywell's Boeing 757 testbed: many pictures with different engines, even turboprops.