Weird Air pump problem

I give up. What model year car are we talking about?

From the 2002 MY Mustang, here is the list of things to check for the P1414 DTC. Since you changed the pump and the relay, my vote is it is a false input from the AIR Monitor circuit. IE, the PCM thinks the AIR pump is running when it is not.

In any case, you need to work through the list of possible causes to rule each one in or out. Start with the grounds. Check all of the electrical connectors. Look for bent/pushed/dirty/damaged pins. Double check all of your work.

Note, there is limited help I can give from here on. From what I can tell, the 2002+ MY does not have an AIR pump. So my reference manauls and wiring diagrams do not have any information on the AIR pump system.

Good luck.
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P1414 Secondary Air Injuction System Monitor Circuit High

The secondary air injection system monitor circuit is high, indicating the electrical AIR pump is on although the electrical AIR pump was commanded off by the PCM.

Open AIR monitor circuit from the pump
AIR circuit short to power
Damaged solid state relay
Open AIR pump ground
Damaged AIR pump
Damaged PCM