Eavesdrop with Beethoven — first half’s organ (6)
I believe the answer is:
earwig
'eavesdrop' is the definition.
(both can mean to listen to someone secretly)
'with beethoven first half's organ' is the wordplay.
I cannot really see how this works, but
'with' could be 'w' (abbreviation) and 'w' is present in the answer.
'first' could be 'i' (I in Roman numerals eg Elizabeth I) and 'i' is located in the answer.
'organ' could be 'ear' (the ear is an organ) and 'ear' is present in the answer.
A single letter 'g' remains which might be clued in a way I don't understand.
This explanation may well be incorrect...
(Other definitions for earwig that I've seen before include "An insect; to eavesdrop (slang)" , "Small insect with aural toupee?" , "Insect with pincer-like tail" , "Clawed insect" , "Close acquaintance" .)