I have the same type of nest box, but the front opens instead of the top. There is enough overhang that water does not run back along the underside of the top. Even though it does not close real tightly, I have never had a problem with water getting inside. Maybe you could rework yours so that the front opens instead of the top. Also, there is a good roof overhang that protects the place where the box joins the house.
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