You remind me of myself when I was starting out with just five chickens and a run made from steel hog panels. I placed hog panels over the top with chicken wire over that to keep predators out. Very slap-dash and rudimentary.
Then the winds came. And the rain. And the snow, and I was trying to fix issues on the fly, or more correctly, being blown away by 50mph winds. Try rescuing a plastic sheeting flapping like a sail in that degree of wind. Ain't the least bit enjoyable.
So, I added fiberglass corrugated panels over the top instead of plastic sheeting. Since the pitch was flat like your run, the snow load wouldn't shed and the roof leaked, too. I added vertical supports to shore up the roof to keep it from collapsing onto my chickens and killing them. This wasn't anything close to being safe.
Finally, I said enough. I hired a carpenter to work with me to build a new run. The proper way. I scavenged materials both free and used and new, but the run was built properly. It's been a joy ever since.
I am saying that short cuts never work. Cheapness at the expense of sturdiness won't work. It's wasted effort and will need tearing down and doing over. Unless you plan on chicken keeping being a short term hobby and then moving on to another fun hobby, do this right.
There are only a few points on your roof supports where the fasteners need removing so you can shim one end creating a pitch so your snow and rain will shed. DO IT!! Then bite the bullet and spend the money on some roofing panels that will go up in just an hour or two and will make a proper roof for your run that won't collapse or leak. And it's done. You won't have to do it all over again at some point, which usually comes at a time when you have the least amount of time to get it done.