Yesterday: I've got the P0410 on my 01 SL1 that I can't solve this time. My pump isn't running upon startup. I checked and have power to the pump upon startup. Usually if the pump has power and doesn't run, then the pump is bad. To make sure the pump is bad, I wired it straight to the battery. The pump runs like it should. I can't figure out why the pump won't run if it has power and is not defective.
Today Additional info. I have a working pump from a previous SW that I saved. I swapped pumps and still have the problem. I also hooked up my multimeter to check for power again. I have power, it stayed on for about 30 seconds and then shut off, so the relay appears to be working.

 

You've made the assumption that the electrical connection is stable and unchanging when you swap one pump for the other.
I'm thinking there is a grounding issue or an issue with the relay and or its wiring such that The positive lead may be damaged and not providing sufficient current to ruin the pump.

Here is a great video about the whole secondary air injection system from some guy in Massachusetts.

 

Look at the ground splices. I have opened them and cleaned corrosion to restore proper grounds. The ground splices have a plastic cover over the six to eight individual ground wires. The ground splice I had problems with was in front of the battery just behind the radiator core support. I had lost cooling fans and the horn on my car.

 

Follow this link and look through this online manual. The grounds on the S series are very similar to my L series car. My air injection pump is grounded in the same splice as the cooling fans and horn.

 

I've had a few other issues I've had to deal with, so I had to put this aside for a while. I've gone through all the wiring twice more, checking voltage and resistance on all. As far as I can find, the ground is solid and I can find no shorts in any wires. I replaced the relay to no avail. I'm totally flummoxed.

 

You need to load test the circuits to find the problem. Static meter readings are only valid if the test fails. They can pass a standard test and still be bad without a load to expose the problem.