Welcome to BYC.
Putting your general location into your profile will help when you ask questions like this where climate matters. You said "Tennessee" but adding east, west, or central would be helpful -- I presume it's not the mountains since you don't get winter.
Personally, I'd remove that hardware cloth on the ground ASAP. It's hard on their feet and the larger they get the less likely the poop is to pass through it.
Additionally, rinsing the poop out that way creates a favorable environment to grow bacteria -- causing odor and risking disease. The general rule of thumb is "dry chickens are healthy chickens".
What you need in your run -- after they've enjoyed destroying all the grass -- is some kind of dry organic material. Straw, pine straw, wood chips, wood shavings, dry fall leaves, other yard waste, etc. -- whatever is abundant and reasonably-priced in your area.
This material composts together with the poop and is, when well-managed, absolutely odor-free.
Heat, not cold, will be your problem and, unfortunately, that coop is very ill-designed to cope with heat.
It doesn't have enough ventilation for even a northern climate, much less the Steamy Southeast.
Here is my article on hot climate chicken keeping: https://www.backyardchickens.com/articles/hot-climate-chicken-housing-and-care.77263/
And my article on coop ventilation: https://www.backyardchickens.com/articles/repecka-illustrates-coop-ventilation.77659/
Sometimes people turn their coop-and-run combo prefabs into just coop -- possibly a good idea for you since you already have an additional run. Here's a thread about this: https://www.backyardchickens.com/threads/my-renovated-prefab-coop.1440258/
Putting your general location into your profile will help when you ask questions like this where climate matters. You said "Tennessee" but adding east, west, or central would be helpful -- I presume it's not the mountains since you don't get winter.

I lined the bottom of the smaller run with 1/4" hardware cloth and have been rinsing the majority of their poop every 2-3 days through the holes. Not certain if this is feasible in the long run though. Should I be putting something down?
Personally, I'd remove that hardware cloth on the ground ASAP. It's hard on their feet and the larger they get the less likely the poop is to pass through it.
Additionally, rinsing the poop out that way creates a favorable environment to grow bacteria -- causing odor and risking disease. The general rule of thumb is "dry chickens are healthy chickens".
What you need in your run -- after they've enjoyed destroying all the grass -- is some kind of dry organic material. Straw, pine straw, wood chips, wood shavings, dry fall leaves, other yard waste, etc. -- whatever is abundant and reasonably-priced in your area.
This material composts together with the poop and is, when well-managed, absolutely odor-free.

We are in Tennessee with little real winter and others here have said they haven't needed anything to heat the coops. We have small ventilation holes as noted on both sides of the coop...would this be enough?
Heat, not cold, will be your problem and, unfortunately, that coop is very ill-designed to cope with heat.

It doesn't have enough ventilation for even a northern climate, much less the Steamy Southeast.
Here is my article on hot climate chicken keeping: https://www.backyardchickens.com/articles/hot-climate-chicken-housing-and-care.77263/
And my article on coop ventilation: https://www.backyardchickens.com/articles/repecka-illustrates-coop-ventilation.77659/
Sometimes people turn their coop-and-run combo prefabs into just coop -- possibly a good idea for you since you already have an additional run. Here's a thread about this: https://www.backyardchickens.com/threads/my-renovated-prefab-coop.1440258/