An engagement for Joe at Gateshead (3)
I believe the answer is:
gig
'an engagement' is the definition.
(gig is a kind of engagement)
'joe at gateshead' is the wordplay.
'joe' becomes 'gi' (GI Joe).
'at' means one lot of letters go next to another.
'gateshead' becomes 'g' (head letter of 'gates').
'gi'+'g'='GIG'
'for' acts as a link.
(Other definitions for gig that I've seen before include "Live performance; light two-wheeled carriage" , "Horse carriage - performer's single booking" , "A palindromic single performance" , "A one-night musical performance" , "A single musical performance - it's a palindrome" .)