Stay after passport centre makes a bit of a mess (7)
I believe the answer is:
splodge
'a bit of a mess' is the definition.
(blob or splodge of mess)
'stay after passport centre' is the wordplay.
'stay' becomes 'lodge' (I am not sure about this - if you are sure you should believe this answer much more).
'after' says to put letters next to each other.
'centre' says to take the centre.
The middle of 'passport' is 'sp'.
'lodge' after 'sp' is 'SPLODGE'.
'makes' is the link.
Can you help me to learn more?
(Other definitions for splodge that I've seen before include "Spot, splash, smear" , "Irregular blot" , "big splash" , "Dirty mark" , "Messy spot" .)