I dont have good news for you.
Looking at the codes I think a #42 is: "Automatic Shut-Down (ASD) relay circuit open or shorted". That could easily be a short or a broken wire buried in that wiring harness, unforturnately.
I defnitely would not describe repairing a bad/melted/corroded engine wiring harness as a 'simple' repair on a Neon. I myself would NOT try to fix one until I had removed it from the car, and I regard that as an unplesant chore of itself. Working in cold weather makes it worse.
But if someone pulls the engine out for any other reason, the engine wiring harness comes with it, and is then easy to remove. Fixing it is still a good bit of drudge work though.