Promise advance payment - argue hopelessly about that (9)
I believe the answer is:
guarantee
'promise' is the definition.
(guaranteeing is a kind of promising)
'advance payment argue hopelessly about that' is the wordplay.
'advance payment' becomes 'ante' (I've seen this before).
'hopelessly' indicates anagramming the letters.
'about that' indicates putting letters inside.
'argue' is an anagram of 'guare'.
'ante' going inside 'guare' is 'GUARANTEE'.
(Other definitions for guarantee that I've seen before include "Formal promise of action" , "Something ensuring a particular outcome" , "Warranty, pledge" , "Written assurance, pledge" , "Undertake" .)