Tiny bit of housekeeping:
Cart is a 2-wheeled vehicle
Carriage/buggy/wagon is a .4-wheeled vehicle.
Anyway…4-wheels will be easier on your Haffie. The only weight on his back is the few pounds the shafts weigh. You don’t have to worry about balancing the cart in the tugs, the ride will be better too! (think wheelbarrow vs. wagon).
The draft is usually lower on a 4-wheeler so the pulling geometry is better for the horse.
Passengers…you can carry them easier…and there’s an amazing number of 4-wheelers out there, much more limited selection in cart styles without going antique…even then, not so many choices.
You can find lightweight carriages and they typically have at least rear brakes. You don’t stop the cart with the brakes, but they do help relieve him of the push in his breeching when going downhill making it easier for him (and you can go faster downhill if you’re compteting.)
No wire wheels…only wood or steel. Wire wheels are not meant to deal with side loading…think of a bike wheel, it’s very strong pushing down on the tire/rim. When you push on the side of the wheel on the hub, you can see the wheel flexing.
If you are going to compete at all in anything besides the most basic “baby classes” you won’t be able to use pneumatic tires and these are the types of tires you see mounted on wire rims (they come off the rim and are considered unsafe for most competitions). The ADS rulebook, which even if shows aren’t doing a rated show tend to follow, says:
Chapter 10 “The Vehicle”, Article 21 (2): "Wire wheeled and pneumatic tired vehicles mayb be permitted in the following classes at management’s discretion (and a lot of shows don’t include them for safety reasons except for mini-horse classes):
a. If it is the show’s first year as an ADS recognized competition.
b. In all pleasure driving classes if the vehicle is an antique wire-wheel vehicle (long wire spokes, hard rubber tires).
c.In maiden, novice or junior pleasue driving classes.
d. In trainng Level combined driving events.
Go for a 4-wheeler. Make sure it’s a cut-under for safety. Shop around a lot, you can find decent prices right now as people sell their toys.