Are these pics of your coop?
If so, I think I see where the untreated plywood roof of the run is giving you problems. It isn't a ventilation issue (unless you have wrapped your run for winter or something) it looks like it's just an untreated and wet wood in a rainy environment issue. I'd put a tarp over it to keep the rain off it and prevent water from soaking into the wood. If you could erect a shelter so the tarp doesn't sit directly on the run roof then that would be even better because it would allow air circulation and the wood would dry faster.
As soon as the wood dried out I'd treat it with a mildicide and paint it with a good exterior paint. KilZ brand makes a good one. Then I'd install some roofing material. Maybe shingles to match the prefab coop you've got? Or poly or metal roof panels. You'll need to address the area where your rooflines meet. That will be a problem spot as rain will infiltrate the seam between the original run and your extension. If you could tie the two roofs together with flashing perhaps or maybe just one continuous roof over the whole thing then you'd eliminate any place that water could seep in.