Incite one to wander round with a horse (3,2)

I believe the answer is:
egg on
'incite' is the definition.
(synonyms)
'one to wander round with a horse' is the wordplay.
I cannot quite see how this works, but
'round' could be 'o' (round can mean a round object) and 'o' is found within the answer.
'horse' could be 'gg' (geegee or GG is a childish term for a horse) and 'gg' is found in the answer.
The remaining letters 'en' is a valid word which might be clued in a way I don't see.
This may be the basis of the clue (or it may be nonsense).
Can you help me to learn more?
(Other definitions for egg on that I've seen before include "Encourage to act" , "Push" , "Soldier may have this" , "Urge (someone) to take risks" , "Prod" .)
