If you cover the run it will stop sunlight getting in when the sun is directly overhead. At other times sunlight will come in from the sides. There are times my covered run provides no shade in the run.
If you cover the run it will stop rain and snow coming in when there is no wind and the rain or snow is falling straight down. If there is any wind rain or snow will blow in from the side. Still, slope the run cover so rainwater run-off flows away from the coop/run, not toward them.
Don't worry about a cover on the run blocking all sun from the chickens, that will not happen. It also will not keep it perfectly dry.
If you cover the run it will stop rain and snow coming in when there is no wind and the rain or snow is falling straight down. If there is any wind rain or snow will blow in from the side. Still, slope the run cover so rainwater run-off flows away from the coop/run, not toward them.
Don't worry about a cover on the run blocking all sun from the chickens, that will not happen. It also will not keep it perfectly dry.