Design probes to check temperature, leading to a prolonged dispute (8)
I believe the answer is:
vendetta
'a prolonged dispute' is the definition.
'design probes to check temperature leading' is the wordplay.
'design' becomes 'end' (both can mean a purpose or goal).
'probes' is an insertion indicator.
'to check' becomes 'vet' (I've seen this before).
'temperature leading' becomes 'ta' (I can't explain this - if you can you should believe this answer much more).
'end' put inside 'vet' is 'vendet'.
'vendet'+'ta'='VENDETTA'
'to' acts as a link.
(Other definitions for vendetta that I've seen before include "Bitter feud between families" , "A bitter Italian feud" , "A prolonged feud of Sicilian origin" , "A prolonged feud, associated with Sicily" , "intense feud" .)