Thanks a lot, but it's not very much! (3)
I believe the answer is:
tad
'it's not very much' is the definition.
(I've seen this in another clue)
'thanks a lot' is the wordplay.
'thanks' becomes 'ta' (UK informal version of 'thank you').
'a lot' becomes 'd' ().
'ta'+'d'='TAD'
'but' acts as a link.
(Other definitions for tad that I've seen before include "Small bit" , "Smidgen" , "little boy" , "Little bit" , "Small amount or degree" .)