Tarp and sand.
Do you have access to wood chips?
I'm not convinced that wet and damp really is that bad for the chickens. I live in one of the wettest places in the world, outside of the equatorial rain forests, in the coastal rain forests of SE Alaska. So I know about dealing with wet. My chickens don't seem to mind the fact that it is always and I do mean always wet here. The ground never dries out. There are standing puddles much of the time. My chickens seem to prefer these areas. They would rather drink from a mud puddle than their waterer. The damp areas seem to get the heaviest work over. I think because the damp soil and damp decaying wood mulch attracts the most insects. Having said that there are some higher areas that are a bit drier and the sunniest of those is their preferred napping spot. They also have a small covered area that they can retreat to when it gets really wet. They will retreat to this dry area when its pouring rain but most of the time they're out rain or not.
My run is mostly pine needle forest duff and wood chips. Wood chips soak up and amazing about of moisture. Bark mulch would likely not soak up as much as wood chips but I would think would still work well. I've never had the wood mold. It does decay and in areas where it's thick you can dig down and find a mycorrhizal layer. But I think this is a good thing. The chickens spend a lot of time scratching in these areas as they have more insect life. All the insect and fungal life make for a healthier soil and more critters to break down and process the chicken waste. All of which makes for a healthier, cleaner, less smelly run overall.
I haven't ever used sand and I suspect it can work well in some climates. Depending upon your area you might need an awful lot of sand to make any difference. Just spreading a little will only result in it being quickly worked into the under laying soil and then you'll just have slightly sandier mud holes. You still need drainage as the sand just allows the rain water to filter down, it still need to ultimately run off somewhere. And that run off will be full of smelly chicken waste so keep that in mind. That can be a plus if you funnel it to feed a garden bed.
If you can't fill the entire run you might consider trying to raise the level of one area (whether you use sand, gravel, or woodchips) so that the chickens have at least some dry areas to retreat too. If you can add some cover even better. They may still enjoy the wet muddy areas but they'll have a place to get dried out when they need/want too.
I have little personal experience with woodchip bark, and associated molds.
I have much experience, both personal and researched, with wood chips, those that come from tree trimmings.
I would venture to say tens of thousands use wood chips as a base for their outdoor runs (deep litter) with great success.