Enter Gateshead on the way out (5)
I believe the answer is:
going
'on the way out' is the definition.
(I've seen this before)
'enter gateshead' is the wordplay.
'enter' becomes 'go in' (synonyms).
'gateshead' becomes 'g' (head letter of 'gates').
'goin'+'g'='GOING'
(Other definitions for going that I've seen before include "factor in horse's performance?" , "State of ground (racing)" , "Departure; current (rate)" , "Running; leaving" , "in action" .)