Did you check to see if there is power getting to the pump? Did you check the air pump fuse? Do you have any power at the relay that you replaced? Is there any power at the combination valve? Is the computer able to ground the relay? Is the computer able to open the combination valve? Did the relay check bad? If not, why did you replace it?
The only way to effectively check the entire air pump system is with a scanner. You can perform continuity tests on every wire, but you cannot verify proper operation without a scanner. You must command the air pump on with the scanner and check to see if power is getting to the pump. Without the scanner there is no way to tell if the computer is even able to operate the relay so the pump can get the power it needs. There is most likely a broken wire or bad connector somewhere, but the only way to find this without a scanner would be to physically inspect every wire in the system manually and just hope and pray that the computer is actually sending the signal to the relay. (You will not know if it is a computer malfunction unless you physically eliminate EVERYTHING else.)
DTC P0410 sets when the PCM commands the air pump on and the combination valve open and there is no change in the oxygen sensor voltage. The problem can be in the air pump circuits, the combination valve circuits, The combination valve itself, the air pump itelf (not likely if you can get it to run with a jumper) a faulty PCM, or even rarely but possibly, clogged-up ports in the exhaust manifold. Again, a scanner is required to verify the operation of the circuits as well as to verify that the problem is fixed after repairs are made.
Please also review this article:
What Else Could Be Wrong?
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