Baby has right to jog (4)
I believe the answer is:
trot
'to jog' is the definition.
(synonyms)
'baby has right' is the wordplay.
'baby' becomes 'tot' (I've seen this in another clue).
'has' indicates putting letters inside.
'right' becomes 'r' (common abbreviation).
'tot' going around 'r' is 'TROT'.
(Other definitions for trot that I've seen before include "Red" , "equestrians sometimes associate with rising" , "Left wing" , "(Horse's) moderate pace" , "Horse's gait faster than a walk" .)