This should not be happening. I don't know what type of supply the TRX uses, linear or switch mode. But with a linear supply unless it has a noisy rectifier, which is very rare, it should not make even the slightest radio noise. A switch mode supply however makes tons of it. Proper filtering can eliminate most and even all of the noise. Poor supplies however are notorious for abusing radio reception. A linear supply however can couple radio noise already on your power system through to the attached radio. In either case running on battery disconnects your radio from this interfering signal.
Any number of things today can cause very severe radio interference, washing machines (electronic controllers on modern models), furnace, computers, monitors, TV sets (a plasma TV generates outrageous radio energy), battery chargers, digital picture frames, coffee makers, printers, home theater receivers, some LED and CFL lamps (they have little power supplies inside), and on and on and on. Anything with a switch mode power supply or computing logic device, CPU, microcontroller, is a potential source of radio interference which these days is nearly every electrical item.