Duckworth
Songster
It looks like OP's duck coop already has a solid surface roof, so tarping the chain link on the front and the gate would not catch a rain or snow load. I can't tell from the picture whether there is a solid wall, chain link wall, or no wall separating the end where the dogloo is from the rest of the leanto. If there is a chain link wall, I would tarp it most of the way up and leave the top open for ventilation. If it is solid, I would leave part of the upper front wall open for ventilation--assuming there is not other adequate ventilation. If there is no wall, I would put up a temporary wall using welded wire or chain link or anything else you have and tarp it as described.A word of caution of I may.....
Your set up looks great!
Putting a tarp or plastic on top of the run will collect snow and rain. Both of those will add a tremendous amount of weight and destroy the run wire, the wood and be a big hazard to walk under.
I don't know what kind of predator pressure you have in the winter in your area, but that is a big factor for us where we live and the above suggestion does not take into account what would be necessary to protect ducks in my location from predators.