Should I close the gaps in the roof with the hardware cloth as well? And the roof does open but it’s pretty heavy, I’d say around 10 pounds.Welcome!
Chicken wire will keep your birds in,so any predator will have an easier time getting them all!
It's totally useless as predator protection, a waste of time and money.
You need something with no openings larger than 1/2" diameter anywhere, including around doorways and at the roof line. Hardware cloth, securely fastened, everywhere!
It also pays to have 2"x 4" woven wire over that hardware cloth, also well fastened.
Use 1" fence staples, close together, into the framing, and then covered with 1"x 4" boards screwed into the framing.
I think I'm seeing larger openings at your roof, and this all needs to be better protected. If that roof lifts up, make sure it's latched with something a child couldn't open, like a padlock.
Raccoons are a huge threat, but not the only problem that can arrive. Think about two big hungry dogs, and rats, and foxes, coyotes, and weasels. Many critters live very well in urban environments!
Take it from somebody that has 2 pet coons. They will just laugh at chicken wire. You can put an electric fence around the outside but with that ivy covered (guessing fence) they will just go over that and get on the roof and go in. The pen is way too close to whatever that it. Your talking about an animal that can scale a perfectly vertical wall up and back down it never missing a lick. A coon can easily lift 10 lbs. I've seen my half grown girls roll a cinder block around. The net says a cinder block weigh about 30 lbs.
I'd suggest cutting your losses and just redoing the fencing with something designed to keep animals out and not only birds in.