Irritated seeing rat outside shopping centre (8)
'irritated' is the definition.
'rat outside shopping centre' is the wordplay.
'rat' becomes 'sing' (both can mean to inform on someone).
'outside' is an insertion indicator.
'shopping centre' becomes 'mart' (large shopping complex).
'sing' placed around 'mart' is 'SMARTING'.
'seeing' acts as a link.
(Other definitions for smarting that I've seen before include "Painful", "suffering from sting", "feeling the pain".)