Why does the service engine soon light keep coming on after I've repaired what it said was needed?

Because it was miss diagnosed the most common reason i have seen for the air injection codes have been a bad combo valve.What are the exact code numbers do you have not code descriptions?Also when you start it cold can you hear the air pump kick on?

The code is p0410. First, we tried replacing air pump with used part. When the used part was on, we did hear the fan run at start up (never heard that before that I can remember and have owned the car for 11years). Mechanic said it was no good. Replaced it again with new part. Light stayed off for 2 days. When came back on, mechanic said a relay was backwards. Light stayed off for 24 hours. Back on again. Took it to autozone...same code. Thanks for your help.

I doubt the relay was backwards those air pumps dont fail often that i have seen.See if you can rent a hand vaccuum pump we can test the combo valve vaccuum source to the combo valve etc.You could have clogged up air injection ports in the head exhaust manifold etc.Lets do this pop off the hard metal tube that goes to the exhaust manifold.Then do a cold start and see if the air blows out of the combo valve that is located on top of the ac compressor.There should be a lot of air coming out of it.

We can't get the top to come off. Is this the right part that we're supposed to be trying to open (picture attached)?

I didnt tell you to remove the top of the combo valve but that is the correc part.Also did you remove the vacuum hose that goes from the solenoid to the top of the combo valve?If not thats one of your problems its missing.What i was asking you to do is remove the steel tube that runs from the combo valve to the exhaust manifold then do a cold start.See if lots of air comes out of the combo valve.Also see if the steel tube is clogged or restricted.Does your car burn oil?

My husband removed the vacuum hose when he was trying to remove the top of the combo valve (which he was never able to) - so no, it is not missing. Is this the tube that we should remove (pic attached)? Yes, the car burns oil, but not a lot.

Yes thats the tube.