<<<I understand that it's NOT AN AUDACITY problem but a hardware problem.>>>
That's OK. We do hardware.
Your power problems may be more connected than you think. You can get buzz problems like this if the quality of the AC power and the ground system is suffering, or you have two different pieces of sound equipment and the power is slightly different between them.
<<<The external power supply may introduce "ripple" onto the supply voltage for every component in your laptop. >>>
Totally. However, that's not the only failure pathway. One of the two prongs of standard wall power is supposed to be "neutral" or power return. If there is some damage inside the wall or outlet, there can be some power on the neutral connection. This puts a small amount of AC power on all the wires of the power supply and the case of your computer.
[Scene cuts to the microphone] The performance is taking place on a metal desk with the microphone in a metal stand. Chances are, the metal desk really is neutral and might even be connected to ground if it's part of the building.
This means the shield on the microphone cable is grounded at the microphone, but connected to a small amount of AC power at the computer--remember, the whole computer, case and all has a small amount of power on it. In engineering we call this a sneak path--interference coming from somewhere you're not expecting.
You already know what the desperation method is, float the computer. You can also float everything but the computer. That would work, too.
If the computer power supply can be plugged into the wall either direction, flip it over. I have one laptop that will flip, but only works really well one way and not the other.
I know nothing about your AC power, but in the US we can buy a simple tester with three lights, to tell us whether our three-prong wall outlets are wired right.http://www.acehardware.com/product/index.jsp?productId=3099774
The correct two lights have to come on.
You can try to filter or isolate your equipment from the AC outlet (they make special expensive transformers for that) or you can try and arrange for everything in the sound system to be plugged into the same wall outlet. You can get this problem if half of your sound amplifiers are connected to the north building power and the other half on the south.