I would remove the chicken wire from the bottom of the coop and replace it with 1/2" hardware cloth. Install HC over the openings between the rafters. You can add a piece of trim board to help reinforce it and cover the cut edge.
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You need some drip edge under your roof panels and some aluminum fascia to cover your sub-fascia so it doesn't rot. I don't see any purlins supporting your roof. You typically need purlins every 2 feet to support tin or polycarbonate roofing.
I'd also open up a top hinged window in the back over the nest boxes. The more ventilation you can have the better. Those nice roof over hangs will allow you to do that and leave the windows wide open even during heavy rain.