I think you've got the MINI FC codes. You need a code reader that can read the P code. You'll want to purchase a Bentley Manual as it has all the MINI FC codes with their associated P code. It's just a lot to look through if you don't have the P code - as it's in order by P code not MINI FC code.
From what I found:
2845 DME Vanos Exhaust Actuator Movement
Try cleaning your VANOS Solenoids - info here:
MINI Cooper R56 VANOS Solenoid Replacement (2007-2011) | Pelican Parts DIY Maintenance Article
0x277D is P0302 Cylinder 2 Misfire Detected
0x2781 is P0300 Cylinder Misfire, Several Cylinders
Bentley Manual where these codes came from:
Mini Cooper, Cooper S, and Clubman (R55/R56/R57) - Books & Technical Documentation - Page 1.
When was the last time you changed your spark plugs? Also, I have seen the 2845 code associated with timing being off and may be the issue due to the associated misfires. Do all the correct diagnostic work before assuming a timing issue. Clean the VANOS, inspect plugs (change if it's been a while). Plug DIY and info here:
MINI Cooper Spark Plug and Cable Replacement (R50/R52/R53 2001-2006) | Pelican Parts DIY Maintenance Article
Let me know what you find out.