Government department do: couple in the end have left (5)

'government department' is the definition.
(Department for Environment, Food and Rural Affairs)

'the end have left' is the wordplay.
'the' becomes 'definitearticle'.
'end' suggests removing the centre (I've seen 'ends' mean this).
'have left' becomes 'fra' (I can't explain this - if you can you should believe this answer much more).
'definitearticle' with its middle taken out is 'de'.
'de'+'fra'='DEFRA'

'do couple in' is the link.
This may not be correct. Some or all of it may be part of another bit of the clue.