Faculty cut, oh dear (4)
I believe the answer is:
sigh
'oh dear' is the definition.
'faculty cut' is the wordplay.
'faculty' becomes 'sight' (sight is a kind of faculty).
'cut' means to remove the last letter.
'sight' with its final letter taken off is 'SIGH'.
(Other definitions for sigh that I've seen before include "Exhale audibly as expression of regret, relief etc" , "Vocal expression of weariness" , "Utterance made by exhaling audibly" , "Deep breath expressing regret, weariness etc" , "Sound of regret" .)