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Is it likely to be the main power (as opposed to ignition voltage),


Yes, since the ignition voltage line is just used as a "sense" line (assuming the main power is wired correctly).


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and, if so, how should I correct that? A better run for the power than coming through the stereo plug where I assume it's coming from now?


That might help a bit, but just so you know, having the battery icon appear at crank-time is normal and doesn't necessarily mean there's a problem. It's normal for a car's voltage to drop during cranking. The question is whether the player's (or kernel's, or PIC's, whichever it turns out to be) sleep/wake behavior is programmed correctly to deal with that situation.

But if you wanted to see if you could improve the voltage to the player, I'd start by examining and rethinking the way the main power run to the player and the amplifier happen. If the amplifier is getting its power from a cable straight from the battery, you could try doing the same for the empeg itself. Always through a proper weatherproof fuse block as close to the battery itself as possible, of course.
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Tony Fabris