"Radio comes on when a door is opened (no key in ignition) and goes off when door is closed."
*GUESSING: Installation issue - someone installed the power/relay to the wiring that probably powers the door lights. When using the cig lighter it is drawing too much power so the radio cuts due to not getting enough juice I would think. What kind of Radio do you have? I would take the radio totally out and replace it with something else and see what happens. Double check the wires and look for anything that may be fray or touching a ground/metal.
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DrCactus
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