I came across the picture below and saw the hole near the centerline on the left which appears to have some carbon next to it. What is its purpose and where does the carbon come from?
The Auxiliary Power Unit (APU) exhaust
This is answered by the following image (taken from this question, color markings are unrelated to this answer):
The item 72 is what you are looking for and it is labelled "APU exhaust". The black residue is soot from incompletely burned fuel.
The Eurofighter Typhoon can use its APU to start the main engine or to provide power when on the ground. From airvectors.net:
The Eurofighter is also fitted with an auxiliary power unit (APU) for self-starting and ground power.