I can't tell by the picture. Is the existing run permanently attached? Or is it removable?
Either way, to make what you already have more secure, I would replace that chicken wire with hardware cloth for starters. Then you might want to add a welded wire type of fencing to the base of the run. Something with big enough holes to allow the grass to come through, but not let digging critters in. However, don't go too big with those holes, because even a weasel can go through a hole that a mouse could. Check with your local hardware store to see what they carry.
Chicken wire keeps chickens in, but won't keep predators out. It is too flimsy. The hardware cloth is a bit heavier than chicken wire where you are concerned about weight, but it shouldn't add that much more. Do you have wheels on your tractor? Adding wheels can greatly improve mobility.
