AIR pump??? The full name for an AIR pump is "secondary" AIR pump. The reason its "secondary" is because normally it plays no part in the "primary" running of the motor.
The only way I can imagine it would affect running would be if it stayed on all the time and confused the O2 sensor. The AIR pump should shut down after cold running.
Definitely get a second opinion or ask more questions. They have already given a diagnosis, I would ask for an explanation. Remember the secondary AIR pump quits working (when functioning properly) as the engine warms up.
Owner 24 bay BSC
Bosch Master, ASE master L1
26 years MB technician