Heading off anguish, time running out for complaint
I believe the answer is:
earache
'running out for complaint' is the definition.
'earache' can be an answer for 'complaint' (a type of medical complaint). I'm unsure of the 'running out for' bit.
'heading off anguish' is the wordplay.
'heading off' becomes 'ear' (I can't explain this - if you can you should believe this answer much more).
'anguish' becomes 'ache' (I've seen this before).
'ear'+'ache'='EARACHE'
'time' acts as a link.
Can you help me to learn more?
(Other definitions for earache that I've seen before include "complaint to The Listener" , "Pain in an aural appendage" , "result of nagging?" , "Aural complaint" , "pain (from hairdryer treatment?)" .)