Purse lip finally: that hurt (5)
I believe the answer is:
pouch
'purse' is the definition.
(thesaurus)
'lip finally that hurt' is the wordplay.
'finally' says to take the final letters.
'that hurt' becomes 'ouch' (I've seen this before).
The last letter of 'lip' is 'p'.
'p'+'ouch'='POUCH'
(Other definitions for pouch that I've seen before include "Container for tobacco or young" , "joey's home" , "Bag" , "Kangaroo's pocket" , "Marsupial's baby carrier" .)