Re: idle cycle/ lights dim. engine cuts out
The secondary air pump relay is part of the K40 ( RCM ) module, not separate.
Overall, it appears to me that you may have a battery ground cable problem, perhaps loose at the terminal or corrosion on the connection at the fenderwell end ?
Always measure the battery voltage from the engine block to the positive terminal, that is what counts. 13.8 - 14.4 is about normal when running, 12.2 - 12.6 when everything is off.