Mr Livingstone has little time for this county (4)
I believe the answer is:
kent
'this county' is the definition.
(Kent is an example)
'mr livingstone has little time' is the wordplay.
'mr livingstone' becomes 'ken' (I can't explain this - if you can you should give a lot more credence to this answer).
'has' is a charade indicator (letters next to each other).
'little time' becomes 't' (abbreviation for time).
'ken'+'t'='KENT'
'for' acts as a link.
Can you help me to learn more?
(Other definitions for kent that I've seen before include "Lear's faithful Earl" , "County; punting pole" , "Canterbury's county" , "One of the Home Counties" , "Home county" .)