。
Do you ever listen to English programs that are broadcast?
Literal
Be-broadcast-[passive] English [possessive-の] programs [object-を] listen [have-the-experience-ことがあります] [question-か]
~ことがある (with the present-tense of the inner verb) means 'sometimes do X, have the occasion to X' — distinct from ~たことがある (past-tense inner verb = 'have done X before'). 放送される English programs is a relative clause modifier — 'English programs that are broadcast'.