Many a slip by Dr Carter (7)
I believe the answer is:
drayman
'carter' is the definition.
(I know that drayman is a type of carter)
'many a slip by dr' is the wordplay.
'slip' is an anagram indicator.
'by' means one lot of letters go next to another.
'many'+'a'='manya'
'manya' with letters rearranged gives 'ayman'.
'ayman' after 'dr' is 'DRAYMAN'.
(Other definitions for drayman that I've seen before include "Horse-drawn cart driver" , "Carter" .)