One grid broken and wearing away (7)
I believe the answer is:
eroding
'wearing away' is the definition.
(synonyms)
'one grid broken' is the wordplay.
'broken' is an anagram indicator.
'one'+'grid'='onegrid'
'onegrid' anagrammed gives 'ERODING'.
'and' is the link.
(Other definitions for eroding that I've seen before include "Gradually wearing away, rock say" , "Wearing away with time" , "eating away" , "chipping away" .)