Wear down cockney girl with poem (5)
I believe the answer is:
erode
'wear down' is the definition.
(to erode is to wear down or decay)
'cockney girl with poem' is the wordplay.
'cockney girl' becomes ''er' ('her' - 'her' refers to a girl or woman - said with the 'H' dropped - a cockney accent.).
'with' means one lot of letters go next to another.
'poem' becomes 'ode' (ode is a type of poem).
'er'+'ode'='ERODE'
(Other definitions for erode that I've seen before include "Gradually destroy" , "Wear away with weather" , "'Deteriorate, wear away (5)'" , "Slowly remove" , "Abrade" .)