Secondary pump is powered via relay and 30A fuse from a circuit actuated from the ECM on cold startup from approx 65 seconds and 10-30 seconds on warm startup. Vacuum lines controls a shutoff air solenoid valve to allow air flow from the pump to the block at the exhaust valves.
Ignition coils do not have anything to do with this startup noise. IMO You have several problems.
Oh yeah and to answer your question earlier........
The only potential problems a bad secondary pump can cause that I can think of (and this is stretching it) is
1) often when they start to go bad they get condensation and then water built up in them that could find its way into your motor briefly on startup (dont think this is a big deal though)
2) You are briefly giving off more emissions than probably necessary but then again you are already driving a compact car to do your part right?
3) You won't be able to get a new Tag in most states without a functioning Secondary Air Injection system.
Just becuase your pump whines doesn't mean there is a problem. Most are very annoying and noisy.
Sorry if this is too much info. I just finished my first coffee of the day.