County with part of Leith on its border! (7)
I believe the answer is:
leitrim
'county' is the definition.
(county of Ireland)
'part of leith on its border' is the wordplay.
'part of leith' becomes 'le' (I am not sure about this - if you are sure you should believe this answer much more).
'on' means one lot of letters go next to another.
'its' becomes 'it' (I've seen this in other clues).
'border' becomes 'rim' (both can mean an edge).
'le'+'it'+'rim'='LEITRIM'
'with' is the link.
Can you help me to learn more?
(Other definitions for leitrim that I've seen before include "County of Ireland" , "Limiter of Irish county" , "Could the county rile Tim?" .)