Hasten to leave without pedestrian egghead (8)
I believe the answer is:
expedite
'hasten' is the definition.
('expedite' can be a synonym of 'hasten')
'leave without pedestrian egghead' is the wordplay.
'leave' becomes 'exit' ('exit' can be a synonym of 'leave'**).
'without' indicates putting letters inside ('without' can be similar in meaning to 'outside').
'pedestrian' becomes 'ped'.
'egghead' becomes 'e' (head letter of 'egg').
'exit' enclosing 'ped' is 'expedit'.
'expedit'+'e'='EXPEDITE'
'to' is the link.
(Other definitions for expedite that I've seen before include "Hasten progress" , "Cause to be done more quickly" , "Hasten (something)" , "Assist the progress of" , "Speed up progress (of)" .)