Assuming the jumper is 14 gauge (give or take) .4 Amps is not enough to melt the insulation off of it. Perhaps somewhere downstream a wire is pinched or touching the exhaust or something. Perhaps when the vehicle warms up is when the short occurs??? All just speculation I know, but this is the sort of thing you should be looking for. You should continue monitoring with your amp meter until the problem occurs to see if there's a spike.......I'm guessing it will jump up to 15+ Amps.

You need to take apart the battery fuse box & inspect for corrosion inside. Here's links to a couple of threads where this is covered:

http://www.toyotavantech.com/forum/s...ttery-Fuse-Box
http://www.toyotavantech.com/forum/s...-Previa-LE-AWD

Yes, it's normal for relays to run hot. They have electromagnetic coils inside. Coils consume power and will run hot to the touch.

Based on what you said so far, if I had to guess, I'd say there's a problem with the AM2 circuit inside the battery cable fuse box & this is why there's no power to the EFI fuse position. If this is the case, the jumper wire is being used as a bypass to supply power to this circuit (band-aid). Although not a permanent fix, and one I wouldn't recommend, it should still work.........unless there's a problem downstream. That's why I'm thinking there's an intermittent short somewhere in the EFI circuit. Check for pinched or otherwise damaged wires in & under the vehicle. Tim