i am not an cfm56 expert but
afaik the 5C differs much from the 5B. its the only engine in the family with a 5 stage LPT and has a significant bigger fan.
correct, the -5C2/3/4/P is equipped with 5LPT-stages (instead of 4 at other CFM56 variants) and also with an additional Booster-Section stage (or LPC) so the pressure level (or combustion chamber inlet pressure/temperature) inside the engine is a little bit higher -> higher EGT.
The position of the EGT-Probes should be similar (2nd stage LPT-stator).
IMHO the CFM56 is a little bit to small for the A340 runs always close to the limits (and sometimes exceeding those), so the engines deteriorate faster and the turbine-efficiency decreases which results in a higher EGT.