Almost running behind time, through this country (6)
I believe the answer is:
latvia
'this country' is the definition.
(Latvia is a European country)
'almost running behind time through' is the wordplay.
'almost' means to remove the last letter (most of the word but not all of it).
'running behind time' becomes 'late' (I've seen this before).
'through' becomes 'via' (via means going through).
'late' with its last letter taken off is 'lat'.
'lat'+'via'='LATVIA'
(Other definitions for latvia that I've seen before include "Country in northern Europe" , "Baltic country" , "Northeast European country" , "Baltic state, capital Riga." , "Baltic republic" .)