I am thinking either:
A) The pedal/cable is hitting something
B) The Neutral Safety Switch on the clutch being triggered is causing issues.
What I would do:
Google it (along with your year/model), find forums for your car and see if anyone else has had this issue.
Check around your pedal/cable to see if anything is touching.
Check the grounds on your radio.
Get the wiring diagrams for your car, see if there is any link between the clutch switch and the radio.
Poke around with a test light. Connect it to the positive and negative leads to the radio, hit the clutch. Connect to known good ground, test. known good positive, test. Follow the wires.