Rebellious bit of Galileo bothered by instrument (4)
I believe the answer is:
oboe
'instrument' is the definition.
(oboe is a kind of instrument)
'rebellious bit of galileo bothered' is the wordplay.
'rebellious' shows that the letters should be reversed in order.
'bit of' indicates a hidden word.
'eobo' is hidden within 'galileo bothered'.
'eobo' back-to-front is 'OBOE'.
'by' is the link.
(Other definitions for oboe that I've seen before include "Musical instrument" , "Producer of notes" , "source of notes" , "Orchestral instrument" , "issuer of notes" .)