Won’t you start with the shed? (6)
I believe the answer is:
shanty
'the shed?' is the definition.
(I've seen this before)
'won't you start with' is the wordplay.
'won't' becomes 'shan't' (I've seen this in another clue).
'start with' says to take the initial letters.
The initial letter of 'you' is 'y'.
'shant'+'y'='SHANTY'
(Other definitions for shanty that I've seen before include "folk song" , "Number at sea" , "sea air" , "A sailor's sea song, or a rundown hut" , "shed" .)