I hope you can post a picture of your sloped chicken coop and run. Pictures can help a lot in these situations where you are asking for help. But, let me post a Google picture of what I imagine you are dealing with and add a few comments that I hope might be helpful.
Here is a Google picture of a coop and run on a slope...
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Let me state that I converted my entire chicken run into a chicken run composting system. I will dump loads of leaves, grass clippings, wood chips, etc... and the chickens will level everything out. Having said that, my ground is level to begin with.
However, in your case, if anything like the picture above, I would suggest doing something the same. If you started dumping compostable litter into the run, the chickens would naturally level it out for you. Over time, the downside hill portion of the run would build up and your run will level out naturally. That's just how gravity and chickens would work together. As long as you have that litter confined by wood, cement, or wire, the chicken run litter should level out on it's own. Then you, and your chickens, would have a nice flat run to enjoy.
I don't disagree with all the comments about adding drainage but let me tell you about my experience using chicken run compost litter that is about 18 inches deep. When it rains heavy, the water just runs through my litter and does not cause any problems. It's like a built-in natural drainage field. All those leaves, grass clippings, wood chips, etc... are firm enough for me to walk on top of in then chicken run. Even when wet.
In the picture of the coop and run on a slope, I would imagine the water running down into the chicken run compost litter and naturally draining out the bottom but leaving the run basically level.
Using compostable materials and your chicken run fencing (wire, wood, cement) to confine it would be a way to terrace the run without have to resort to using heavy equipment or shovels to level off the dirt. Just let the compost litter find its own level.
I get more value of my chicken run compost than I do selling our excess eggs. I create and process hundreds of dollars' worth of compost every year in my chicken run that I use in my raised bed gardens. In many ways, your sloped chicken run would be better for making compost than my level chicken run field. You would only have to dump the litter material on the high side of the run and the litter would naturally migrate downslope over time, composting all the time. Then you could easily harvest the finished compost at the bottom of the run. Just let the chickens scratch and peck that litter and move it along with gravity. It's a built-in compost conveyor belt!
If something like making chicken run compost is in your future, I would add some kind of access (door, gate, removable panel) on the bottom of the hill so you can easily harvest the finished compost.

Anyways, I am a flat lander and would love to hear what other people think of my composting idea to level the chicken run idea would work on a slope. I just know that I can build heaps of leaves, grass clippings, or wood chips in my chicken run and the chickens will level everything out in no time. I think it would work the same way on a slope, with more litter building up faster on the downside of the run. You would not have to move soil with heavy machines or manual shovels. That saves a lot of work. The chicken run fencing can act as your terracing barrier.

I think it would work. In any case, I wish you luck. Hope you can post a picture of what your setup looks like and maybe others will have more thoughts for you.