If the "hum" increases in frequency as the engine rpms are increased, that's a pretty good indication that it's alternator or ignition related. You can buy filters, but it's easy enough to make your own following the directions posted here:
The adapter is a DC-DC power inverter which completely isolates the load from the vehicle's electrical system rather like a transformer does with AC. You may discount any likelihood of AC ripple (alternator whine) on the vehicle system, it appears the output current is insufficient to power the receiver properly when the lamps are on and drawing additional current. It's all in the active regulator/filter on the 9V side being overloaded so AC ripple from the inverter circuit is not being entirely smoothed out.
There are two solutions to the problem, use an adapter with sufficient output current capability to power the unit at full load or just put up with the hum when the back light is on. You really don't need it lit all the time now do you?