You might have a bad air injection reaction solenoid. THis is very common. Be sure that is has power to it with the key on and that its getting a signal from the computer to operate. IF this solenoid is disconnected, it will definitely cause this code.
YUP! THats it!
P0412 Secondary Air System Solenoid Circuit Malfunction
Possible Causes AIR solenoid connector is damaged (inspect pins for damage, and moisture)AIR solenoid control circuit is openAIR solenoid control circuit is shorted to ground or shorted to voltageAIR solenoid Ignition circuit is open (check the AIR PMP fuse in junction box)AIR solenoid has failed
Here is a picture of the pump. The solenoid will be near by.