In public argument, hard to defeat (9)
I believe the answer is:
overthrow
'defeat' is the definition.
(I've seen this before)
'in public argument hard' is the wordplay.
'in' indicates putting letters inside.
'public' becomes 'overt' ('overt' is associated in meaning with 'public').
'argument' becomes 'row' (I've seen this before).
'hard' becomes 'h'.
'overt'+'row'='overtrow'
'overtrow' going around 'h' is 'OVERTHROW'.
'to' is the link.
(Other definitions for overthrow that I've seen before include "Vanquish" , "Worst" , "Ruin" , "Extra run perhaps" , "Fall" .)