You need to walnut blast or clean the valves in order to truly get rid of the carbon build up. Unfortunately, this is something that needs to be done about every 50k miles on the R56 N14 direct injection engines. How many miles are on your R56 and have you done a true carbon clean up? We have more info on it here:
MINI Cooper R56 Turbocharged Engine Carbon Cleaning (2007-2011) | Pelican Parts DIY Maintenance Article
What scanner are you using to plug into the OBDII to check for faults in the DME? Are there any fault lights triggered on the dash at all? You'd most likely get misfire and a code that points to the HP if it was faulty, but it could be an issue. You can test it to rule it out of the equation. Figure #10 at this article here:
MINI Cooper R56 Fuel Pump Testing (2007-2011) | Pelican Parts DIY Maintenance Article shows you how to test the HPFP with a scan tool.
Let me know what you find out.