Oh. No horse is going to peddle.
Why don’t you just stick a solar/electric motor on the cart to make the cart easier to pull for the horse? The horse would become less tired.
hybrid would be best, gas or electric as backup. For animal like horse, a treadmill or pedal design to propel vehicle manually
Take a car, add a horse in a cage using pedals, instead of pulling it the horse/cart way, making a huge bulky thing you call a "bike". Yea, you can quit your day job now.