Old poet in middle of school run (6)
I believe the answer is:
horace
'old poet' is the definition.
'middle of school run' is the wordplay.
'middle of' means to look at the middle letters.
'run' becomes 'race' ('race' can be a synonym of 'run'**).
The central letters of 'school' are 'ho'.
'ho'+'race'='HORACE'
'in' acts as a link.
(Other definitions for horace that I've seen before include "- Walpole, eighteenth century English author" , "Roman poet and satirist" , "Roman poet; -- Walpole" , "Boy's name - Roman poet" , "Roman lyric poet, d.8BC -- a chore (anag)" .)