We'd recommend clearing the P0456 trouble code then driving your Mini Cooper at least through 2 drive cycles in order to reset the required emission monitors (which will be cleared when you manually erase the trouble code with your scan tool), and to see whether P0456 will reappear.
Follow the drive cycle below. Insure your Mini has between 1/4 and 3/4 full in the gas tank (the drive cycle will not start of fuel level is not within this range). Perform the drive cycle on public road ways at your own risk. It is highly recommend to perform the drive cycle on a dynamometer rather then on public roadways, as it might be difficult following the drive pattern in traffic.
1. Start engine and idle while cold for 2 minutes and 10 seconds.
2. Accelerate to 20-30 miles per hour and maintain steady speed for 3 minutes and 15 seconds.
3. Accelerate to 40-60 miles per hour and hold steady speed for 15 minutes.
4. Decelerate and come to a complete stop. Idle engine in gear for 5 minutes.
The drive cycle above will be terminated if engine RPM exceeds 3000 at anytime. Or if road speed exceeds 60 mph. Or if there are large fluctuations in throttle angle.
After performing the drive cycle at least twice, check for trouble codes and ensure all emission monitors are complete or "ready". If all is well, proceed to smog check.