First in running — in dream run (10)
I believe the answer is:
administer
'run' is the definition.
(I've seen this before)
'first in running in dream' is the wordplay.
'first' becomes 'ist' (ist resembles est).
'in' is an insertion indicator.
'running' becomes 'admin' (I've seen this before).
'in dream' becomes 'er' (I can't explain this - if you can you should give a lot more credence to this answer).
'admin'+'er'='adminer'
'ist' put within 'adminer' is 'ADMINISTER'.
Can you help me to learn more?
(Other definitions for administer that I've seen before include "man?" , "Organise" , "Deal out (punishment)" , "give" , "act as manager" .)