Anxious to get Livingstone out of the way! (4)
I believe the answer is:
keen
'anxious' is the definition.
(I've seen this before)
'livingstone out of the way' is the wordplay.
'livingstone' becomes 'ken' (Ken Livingstone is a famous UK politician).
'out of' is an insertion indicator.
'the way' becomes 'e' (Compass direction - East).
'ken' placed around 'e' is 'KEEN'.
'to get' acts as a link.
(Other definitions for keen that I've seen before include "Quick" , "Anxious" , "Agog" , "Fervent" , "cry" .)