During the Winter months we try and stay on the cart paths as much as possible. Only in rare instances we will allow carts on the turf. This is because any snow melt or rain that has occurred does not move through the soil profile as fast during cold weather. The ground is generally frozen beneath the surface and therefore does not allow water to percolate through.
Also, any damage caused by cart traffic is not going to be repaired until sometime in the Spring due to the lack of plant growth.