2 of my catios has hardware cloth around the sides and bottom to keep animals from getting inside but the top is regular yard fencing.They are cheap to make.
You may have to attach a wood door frame to the kennel gate to close the gaps .That would definitely improve the security of the run if you're interested.The lessor weasel is very small in comparison to a regular weasel...
I can see you've put a lot of work into this coop and run but don't see an apron around the bottom to keep our diggers. If it doesn't have hardware cloth buried around the bottom coons, skunks,weasels and coyotes can still get inside.