If the LED does not flash but the secondary air injection pump relay clicks:

WARNING: This page does not describe the selected car, but rather 13 other vehicles, including the 2005 Volkswagen Jetta, 2005 Volkswagen GTI, 2004 Volkswagen Jetta, 2004 Volkswagen GTI, and 2003 Volkswagen Jetta. However, it is still accessible from the selected car via links, so may be relevant.

Proceed with Output Diagnostic Test Mode until completed.
Switch ignition off.
Check wire between secondary air injection (AIR) pump motor V101 and contact 1 and body Ground (GND) for open circuit.

If no malfunction in wire is detected:

Fig 1: Secondary Air Pump Relay At Relay Plate In Control Module Protective Housing

Courtesy of VOLKSWAGEN UNITED STATES, INC.

Pull secondary air pump relay - arrow - off relay plate in control module protective housing.

NOTE:

If tools are necessary to pull relays or control modules out of the relay plate, first disconnect battery Ground (GND) strap.
Before disconnecting battery Ground (GND) strap obtain code for radios with anti-theft coding.

Using wiring diagram check wire between secondary air injection (AIR) pump motor V101 contact 2 and contact 2 on relay plate for open circuit.

If no malfunction in wire is detected:

Using wiring diagram check the voltage supply (terminal 30) for secondary air injection (AIR) pump relay J299.

If voltage supply is OK:

Replace secondary air injection (AIR) pump relay J299.
Check DTC Memory, if necessary, repair any malfunctions and then Erase DTC Memory → DTC memory (functions 02 and 05), checking and erasing
Read readiness code → Readiness code, reading . If DTC memory has been erased, readiness code must be generated again:

Engine code AWW, → Readiness code for engine code AWW, generating

Engine code AWP, → Readiness code for engine code AWP, generating