Caught, not many would quarrel at length (4)
I believe the answer is:
feud
'quarrel at length' is the definition.
'caught not many' is the wordplay.
'caught not' becomes 'feu' (I am not sure about this - if you are sure you should believe this answer much more).
'many' becomes 'd'.
'feu'+'d'='FEUD'
'would' is the link.
I am not very happy about this link. It may be part of another bit of the clue.
(Other definitions for feud that I've seen before include "Sustained enmity" , "Private war" , "quarrel without end" , "Long-running disagreement" , "Vendetta" .)