The extent to which many get a bit put out (5)
I believe the answer is:
ambit
'the extent' is the definition.
(ambit is a kind of extent)
'many get a bit put out' is the wordplay.
'many' becomes 'M' (Roman numeral for a thousand).
'get' says to put letters next to each other.
'put out' is an insertion indicator.
'a' next to 'bit' is 'abit'.
'm' placed into 'abit' is 'AMBIT'.
'to which' acts as a link.
(Other definitions for ambit that I've seen before include "Circuit, compass" , "Scope or confines" , "Limits" , "round" , "Range - surroundings" .)