New code-P0410

[smithy]

I recently changed my O2 sensor after my CEL came on and I pulled a OBD11 code for it. Now, a day later, I got the CEL again and a new code-P0410-secondary air injection system malfunction. Through a bit of research I came up a cause- an SAS valve that is stuck open causing water to be allowed into the air pump and exhaust. This makes sense because when I start my car, I get a puff of white smoke indicating moisture in my exhaust. (please, let me know if I'm off base)

Now for my questions. Is there any chance that this code could be caused by a vacuum leak or something other than a faulty valve? How would I check this before changing out the valve? What are the chances of the Air pump already being shot along with the valve? What effect could this have on my gas mileage? (As a side note, my gas mileage is very low right now. It is in the 10-15 MPG range for city traffic.) Any clues with this would be very helpful. Thanks.

[Jot]

On the first start of the day the air pump should run for about two minutes. depending on the model year The relay is attached to the pump but later ones it was moved above the radiatoras it would fail. The white moisture can be condinsation in the exhaust system