AFAIK, the icon movement indicates that Windows is crashing. On reboot, it rebuilds the desktop. But whether the crash is hardware or software related isn't clear. Can you monitor the supply voltages during operation and watch for a voltage sag or dropout when it crashes? Can you replace the power supply?
I don't remember, does XP have an event viewer? Any clues there?
More hardware info is required to give any more help. Make, model, configuration, etc.
Ed