Ideas for bare ground run

I have 12 hens in a run about 30'x75' for the past year-and-a-half and they've scratched down to bare ground. I need to give them more foraging space so I moved one part of the fencing the other day but it's not nearly enough. I need to keep a long tree line enclosed inside the run for shade and shelter from rain. And my other concern IS rain - I'm in West Central Florida hurricane territory and last season it got really muddy in there so I'm trying to be proactive by moving the fencing now before May.
Problem is the area I can move them to is really just crunchy brown weeds and no grass. Florida soil is really just sand and dust. I'd be surprised if there were any bugs in there.
I don't know what to do. I've spent a good couple of months thinking this through and I still haven't come up with a good plan. It's not like I can sow seeds and grow some crop or something for them to eat. Paul talks about the ideal system being one where you can move the fencing in rotation. I can move my fencing, but to where?
Can anyone help? I'm not desperate, but I'm getting there.

Our chicken run has a lot of bare ground and is frozen for at least 4 months out of the year.   Our current solution is a deep layer of wood chips added to the heavy traffic part of the chicken run at least 2 a year.  We can easily get wood chips and they are great source of carbon to absorb the droppings over the winter.  They tend to break down rather quickly so we remove them in the late spring after mud season is over and fresh chips to the run.  

Spring through fall we also add a temporary chicken run with movable fencing.  We have several different configurations so they have periodic access to other parts of the yard.  

A deep layer of wood chips would suit for you the most. But the layer must be thick enough to keep the soil underneath still moist, for the earthworms to be able to crowl and survive there. Deep mulch bedding is a great thing for chickens to scratch and search for small creatures to eat.. Some more fruit bushes (currant, elderberry, mulberry etc.) would help to shade the ground and slow the evaporation of the water from the soil/mulch and would give some more food for the girls. You can throw your kitchen and garden waste there, chickens will eat what they choose and also mix the leftovers with the wood chips to a great compost.  You can empty the partly composted bedding once/twice a year (as Kate Muller mentioned) for your garden.

I have a  backyard bare of annuals because of just 5 chickens.
I pile autumn leaves in the back bed and they spend most of their time there, scratching, even though I feed them up by the house.
So,  definitely give them wood chips, but I would subdivided their yard as well.
A six foot by 75 foot corridor along one side of the run would give access to four yards,  each about 18x24.
Pile your wood chips and compost into the corridor,  grow stuff in the yards, cycle them them through each yard for a day or more.

You could subdivide the yards even smaller,  make each of them 10x 24, and have one for each day of the week.

I would also cover most of  the corridor with tarps,  diverting the rain into the grow yards and into rain barrels.

On the fence lines,  morninga and willow trees,  maybe mulberries.

You could build mobile shields to keep chickens out so the grass grows. 4 2x6s to make a box 2ft x 8ft, then staple fencing on the top. Grass will grow. You can either leave it and they eat what grows through, or you let it get flush with green then move it, giving them full access and you start again. What really blew my mind is the stored seed bank. I seeded it but as i moved it and moved it, grass grew under it without reseeding.

A water trough with duck weed will give them high protien greens. They double every couple of days but the chickens may eat it all. The solution is similar to the grass example.  Cover a portion of the trough with screenwire. What duckweed is under the screen cannot be reached by the chickens.

Consider your stocking rate. Would 6 chickens give enough eggs and do less damage?

You guys are amazing. I want to do all your suggestions, but I'll control myself and start with a few.
I already have 2 spare portable raised beds I built. 4x6'x12"so I'll cart them over to the run today and buy seed. I've got spare cage screens that go over each bed frame that I can use to keep them out till there's good growth. Whoa.
I have a kitty pool that all used for the duckweed. I use those flexible pipe misters to fill the water bowl. Keeps the water fresh. I just pull out the mister and use the pipe because it shapes to the side of the water bowl. Now I just have to locate some duckweed. I know I'll find it around here somewhere. Whoa.
I can move the long section of the fencing closer to the tree line and try to replant the dirt. Leaves are hard to come by here in Florida and I've been keeping an eye out for yard waste. I bought a little chipper shredder a few months back and it's pretty sad but it's something. I've got an awful lot of trees around here. I've also got several rotted wood piles and I can throw those in there. Whoa.
I have 4 foot high poultry wire to hook up for temporary fencing and create sections. A lot of people think I'm crazy because my fencing isn't a fortress. The tree line fence is chain link and while the chickens can't fly out, anything can climb over the top. It isn't to keep the foxes out, it's to keep the chickens in. Varmints are dealt with another way. And in my experience no amount of fencing no matter how sturdy will keep something out if it really wants to get in. The coop on the other hand IS a fortress. Not a mouse or a snake can get in there. I'm home 90% of the time, and the dog keeps any daytime predators away.
I have steady customers for eggs and have to hoard some for myself. They'd be very disappointed if I cut the herd down lol. I really do have enough room for 12 or more birds I'm just having a hard time trying to figure this out. I have a very good mechanical brain but this thing has me in a twizzle. That's why I asked for help because I just can't wrap my mind around it.
Okay, off I go! I've got my work cut out for me today! I can't thank you all enough. But thank you thank you thank you.

Just wanted to post a quick update. My neighboring farmer had a big pile of dead leaves and said I can have it so I take my card over there every other day or so and load it up. There are tons and tons of grubs in there! Bonus!
I put up the baby pool, found some duckweed and just like you said they ate it all. I replenished it twice but it was gone by the end of the day. I may keep a little on the side in a separate pool to give to them a little at a time. It's free, after all.
This is as far as I've gotten but I thought I'd post a pic anyway.

Looks like you're doing fine, Leslie and there's some good suggestions.  If I was in your situation, I'd divide the run into 10 15x15' sections and rotate them through a week at a time.  That would give your paddocks 10 weeks to recover, allowing you to seed what you want.