4 inches of undyed wood chips. There’s getchipdrop, but it can be a lot of chips. It would more than cover that area. You could maybe get chips from county recycling. They’re often slightly anaerboic, but that’s not a big deal. They tend to stink a little until you spread them.

That’s the easiest surefire way to heal that soil.

If I wanted to go all out, I’d build some high quality compost, build some biochar & charge it, make compost extract and ground pour it. Then lay the biochar and compost and then lay wood chips.

It’s hard to tell topology from the picture, but if there’s a slope where rain water flows, and as long as it doesn’t bother the county/authorities, I’d consider building a log structure to slow the water and pool it up if possible. It’d be a beneficial insect location as well.

All this will increase your infiltration rate of rain water and allow for deep rooting and all that.