Trevi...if you're talking of using a tarp to block snow - not a good idea due to the weight of the snow. I do use one to help with rain, but I've punched several strategically placed holes in it so pooling doesn't occur. Next spring, I'd work on drainage around your run. But you can always place some blocks, stumps, etc. in there for them now. Maybe next year you could try gravel w/several inches of sand on top???
And yes, if you want them to come outside, you'll probably need to shovel when it snows. Mine wouldn't touch more than a half inch or so last winter, so I shoveled a lot...big clearings, paths, etc. I figured it was my winter time exercise. I swore I was going to roof part of my run this year, but it hasn't happened.
My winter time routine is: windows closed (pop door always open during daytime hours though), claypot water heater for inside waterer, rubber water bowls outside changed out 2x a day w/warm water, extra scratch, heat lamp near roosts (hung from chain w/backup wire) for severe cold, and thicker bedding. I put tarps around the prevailing wind sides of my run, and add a windblock in front of the pop door.