Detectives overlook everything, nearly giving notice (9)
I believe the answer is:
dismissal
'notice' is the definition.
(I've seen this before)
'detectives overlook everything nearly' is the wordplay.
'detectives' becomes 'dis' (Detective Inspectors, DIs).
'overlook' becomes 'miss' (synonyms).
'everything' becomes 'all' (all things).
'nearly' means to remove the last letter.
'all' with its final letter removed is 'al'.
'dis'+'miss'+'al'='DISMISSAL'
'giving' is the link.
(Other definitions for dismissal that I've seen before include "sending off" , "Rejection" , "brush-off" , "Notice" , "job loss?" .)