Travel around York’s middle part (5)
I believe the answer is:
orbit
'travel around' is the definition.
(I've seen this in another clue)
'york's middle part' is the wordplay.
'middle' means to look at the middle letters.
'part' becomes 'bit' (both mean a piece of something).
The middle letters of 'york' are 'or'.
'or'+'bit'='ORBIT'
(Other definitions for orbit that I've seen before include "Circle - influence" , "Stellar path" , "Go round" , "Planet's path" , "heavenly passage" .)