Pudding a firm one Irishman sent back (7)
I believe the answer is:
tapioca
'pudding' is the definition.
(tapioca pudding is a kind of pudding)
'a firm one irishman sent back' is the wordplay.
'firm' becomes 'co' (a firm is a company).
'one' becomes 'i' (Roman numeral).
'irishman' becomes 'Pat' (Patrick is a common Irish name).
'sent back' says the letters should be written backwards.
'a'+'co'+'i'+'pat'='acoipat'
'acoipat' back-to-front is 'TAPIOCA'.
(Other definitions for tapioca that I've seen before include "Foodstuff used as a thickener" , "Starchy white substance used to make puddings" , "Milk pudding ingredient" , "Starchy food" , "Pudding maker" .)