Winter Run?

Lots of folks cover their runs with clear plastic so that light still gets in but the run is protected from the elements. Ventilation can be maintained by leaving the ends "open" etc. I believe @Blooie is one that has done this so looking at some of her photos may help you see how it works.
Alternatively, you can go out and place a light cover of straw, leaf litter, etc over any fresh snow cover that occurs. I have always had uncovered runs (covered by wire, but not plastic, roof, etc) and managed to keep them out and active by simply making sure I had knocked off any snow that managed to accumulate on top of the run during a snow fall then putting out a couple of flakes of straw shaken around on the top of the snow. My last run was 10x20 and in an area that received frequent snowfall all winter long, I only used 2-3 bales of straw total each winter through this approach.

 

I use a deep bedding in my coop of pine shavings.

My run has deep litter.

About once per year when I remove the soiled deep bedding, instead of tossing directly into the deep litter run, I store it in strategically placed garbage cans with lids to keep it dry.

After a snow event, I toss shovel fulls of the spent litter on the snow covered run.

The birds venture out and spread it nicely and have no further issues with the run.