A sort of exercise — one carried back in a carriage, to begin with (7)
I believe the answer is:
aerobic
'a sort of exercise' is the definition.
'one carried back in a carriage to begin with' is the wordplay.
'one' becomes 'a'.
'carried' becomes 'bore' (I've seen this before).
'back' is a reversal indicator.
'to begin with' says to take the initial letters.
The first letters of 'in carriage' is 'ic'.
'bore' back-to-front is 'erob'.
'a'+'erob'+'ic'='AEROBIC'
(Other definitions for aerobic that I've seen before include "(Of exercise) improving the use of oxygen" , "Exercise that enhances circulation" , "Of a process depending on oxygen" , "Requiring free oxygen to breath" , "(Of exercises) related to increasing oxygen in blood" .)