Any way to divert the water away from the coop/run? Dig a trench, put in a drain, etc?
Otherwise, remove the cement. Prepare the ground properly, then pour cement. But pick the right kind of cement, and pour it thick enough.
But, the cement you have may have cracked simply due to that moisture and freeze/thaw cycles over the years. Therefore, you may want to divert water away in any case.
In the meantime, until you can fix the flooring issue, you may just want to pick a material for now (straw, shavings, etc).
You'll want to fix the cement floor in any case, other posters have had real issues when the rats found them and then burrowed under the cement, and came up into the coop due to the voids and cracking.
Good Luck.