I couldn't find one but did it ever exist? Would adding fuel injection have improved combat effectiveness or fuel efficiency?
Yes there was. A Spitfire which was converted by Daimler Benz engineers to use a DB 605 engine. See the following links:
I couldn't find one but did it ever exist?
I do not think so (at least not off the assembly line) the RR Merlin Engines were Carb'ed. The Griffon powered planes were also carb'ed however they used an injection carb which is a bit different than a float style carb but not full on fuel injection.
Would adding fuel injection have improved combat effectiveness or fuel efficiency?
Yes, it would have allowed sustained G maneuvers which are not possible in a carbureted plane as the float would fall causing fuel to enter the carb unrestricted.
However, the Merlin's float controlled carburettor meant that both Spitfires and Hurricanes were unable to pitch nose down into a steep dive.
This was later remedied by installing a small plate over the float to prevent starvation.
...original production variants of the Merlin used an SU manufactured carburettor in which the fuel flow was metered through a float. In most circumstances this proved to be sufficient but during the air battles over Dunkirk and during the Battle of Britain it was found that whenever the Merlin was subjected to negative "g" forces, such as a quick "bunt" into a dive, the engine would briefly lose power through petrol starvation...
...The remedy, invented by Beatrice "Tilly" Shilling, was to fit a metal diaphragm with a hole in it, across the float chambers. It partly cured the problem of fuel starvation in a dive. The device was commonly referred to as 'Miss Shilling's Orifice'....