ok I have a 2000 s-10 2.2 4 cyl, which had the code, p0410, secondary air injection malfunction. I first was told it was the air pump, so I bought a new one and installed it, had the code cleared, and BAM driving it the next day the light came back on with the same code. Well after doing some research I came up with my own diagnostics.
1. follow the hose going out of the air pump to the top of the engine, disconnect it, and on a COLD START, the air pump should be pumping air through the hose, if it is not then the fuse is blown, or the pump is bad. There is 2 in-line fuses for this system by the battery, it is best to take the battery out to get to them, because they are not in the fuse box. One is a MAX 30A, the other, a typical 10A.
2. If the pump is doing what it is supposed to on a cold start, there is two other probable causes that I know of, one is about a $25 vacuum operated switch, and the other device is some kind of one-way air flow (do-hickey) which costs about $25 dollars also. These all connect on to the air pump one way or another, and the only thing you need to note is the in/out hoses, and replace those cheap hose clamps, with some quality, reusable stainless steel ones.
After replacing the one-way switch on my truck, the light still came on, and the problem all along was the $25 dollar vacuum switch connected to ther air pump. So after a few unneccary, costly repairs my truck didn't need, it is doing fine again. Oh well at least I have a whole new secondary air injection system now, kept the old parts as spares, and can't complain about that being the only thing so far with 70,000 miles on it!!!