Then any suggestions regarding the secondary air injection pump? possible to test it somehow? any fuses?
The pump fuse is the big orange one ( 40 amp ) on the RCM.
Yes, you can test it by connecting +12 to one pin and ground to the other. ( ground to the pin that had the brown wire connected to it )
Most likely a stuck relay in the RCM, sent it to DJ ( tighed1 ) and he can replace the relay for you.
Meanwhile, just remove the air pump fuse, won't hurt anything if the pump don't run, other than passing emissions tests.