Live with petty put down (8)
I believe the answer is:
belittle
'put down' is the definition.
(I've seen this in another clue)
'live with petty' is the wordplay.
'live' becomes 'be' (synonyms).
'with' means one lot of letters go next to another.
'petty' becomes 'little' (synonyms).
'be'+'little'='BELITTLE'
(Other definitions for belittle that I've seen before include "Imply that thing or person is of little value" , "Denigrate, disparage" , "Make small" , "Sell short" , "Imply that person or thing is unimportant" .)