IMHO, assuming the cart path is artificial,it is an obstruction and you are entitled to relief including your stance. If the ball were on the path, it is possible that if you allow for your (right handed) stance the nearest point of relief is on the mown left side of the path: ie. relief may be the edge of the path on the left or a metre or so off the path on the right to allow for your stance. Then the nearest point of relief is the mown left edge of the path. You then have one club length, no nearer the hole.There is nothing in the rules that you can't get a lucky break!

However, if your ball already lies in the rough on the right, allowing for the width of the cart path, I doubt very much if the nearest point of relief could possibly be on the left. You need to take your stance clear of the path and mark where the ball would be. You then have one club length no nearer the hole for your drop.

A bit difficult to write but I hope this helps.