Priest runs after page 3, for example (6)
I believe the answer is:
rector
'priest' is the definition.
(I've seen this before)
'after page 3 for example' is the wordplay.
'after' means one lot of letters go next to another.
'page' becomes 'recto' (right-hand page of an open book).
'3 for example' becomes 'r' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'recto' next to 'r' is 'RECTOR'.
'runs' acts as a link.
This may not be right. It may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for rector that I've seen before include "Minister of religion" , "Vicar" , "Parish clergyman" , "Churchman" , "Member of the clergy" .)