Hoary Cress - What does it mean for my soil?

So in my ~25'x25' "garden", hoary cress is the dominant "pioneer plant" AKA weed. My soil is mostly partially decomposed Mancos shale, with some super-hard (when dry) clay, and some sand a little deeper in places.
I know there's not much organic matter in the soil, and it's pretty poor in general. I've not put fertilizer on it for about 3 years now. That was before permaculture smacked me in the face.
Obviously, I need to add organic matter. In the past I've added llama poo, hay & straw. Last year I had a dismal veggie crop, and now my wife's starting to doubt the efficacy of PC.
I'm probably going to do something like the guy in the Back To Eden movie - backtoedenfilm.com

Really the question is, what does the hoary cress mean for soil indications and does it mine or accumulate nutrients (like comfrey)? Will it go away as the soil gets better?
Also, any suggestions for improving gardening success on the cheap?
Thanks!

Hi Cam,
I just did a search for hoary cress on the plant forum here, (and for whitetop, another name for it (Cardaria draba), but it didn't hit). I am also seeking advice, so I thought I'd bump your question instead of starting a new thread, as our situations are a bit different, but with many similarities, so maybe with both we can get some brain storming going. It's been a while, how has your situation progressed?
I'm helping out a local non-profit that has an EcoGarden, basically a small Toby Hemmenway-esque guilded food forest. It was planted between 6 and 9 years ago and is in decent shape, though it could be better. The management of it has lapsed, as the person who got it going left and the remaining members, while interested, don't have much permaculture experience. They've been removing organic matter from the guilds, and there is a fair amount of bare soil and grass present, so that's the first thing I'll change and get more organic matter in there and get them to institute chop and drop management. However, there is also a pretty serious whitetop infestation. When I worked for the forest service many years ago, we hit whitetop with a concentrated 2,4-D and dicamba mix, which kills broadleaves; obviously not something I'm going to do now. I never learned any other methods of control, and there seems to be sparse data online of ways of modifying the site to make conditions unfavorable to it.
My understanding is that it thrives in salty soil and full sun. I'm in the Salt Lake Valley (Utah), and the site in next to a river on one side and a lawn that gets summer sprinkling on the other, so the site is fairly moist. So a more shade producing canopy would seem to help, however, there is plenty of the whitetop right in the guilds, so it seems to be coping with partial shade just fine. My thinking is that greatly improving organic matter would probably help tip the scales toward the desirable species and away from the whitetop. Also, it sounds like herbaceous legumes can compete with it to some extent, so I'm thinking of getting a dryland legume seed mix (white clover, maybe alfalfa (a bit water hungry), sweetclovers, etc.) planted densely in the guilds for competition.
Whitetop spreads by both seed and running roots, so digging it up, especially in the midst of the foodforest, isn't really a viable possibility. I can chop and drop it to reduce seed production, but that will cause it to send up more shoots. Maybe seeding with a legume mix, followed by chop and drop of the extant plant shoots. Then hopefully the legumes will have a chance to out compete it and shade it out? May take a second selective chop and drop in a couple weeks. Any advice would be appreciated.

Derrick: You're right on in your thinking. Hoary Cress prefers alkaline/salty soil. All the photos I found show it growing in full-sunlight, and usually in drylands. From that I infer that it doesn't compete well with other plants that might shade it or that would out-compete it if there was more clean water available.

Hoary Cress is a perennial, therefore tilling breaks the roots up into lots of small pieces which then proceed to start growing again.

In small beds, digging it out and screening the soil to remove roots is fairly successful... Like any plant, if the above ground portion is kept cut off, or smothered, eventually the root will die. Mowing just before the plant flowers greatly reduces the seed load in future years.

Cam: To me gardening on the cheap means growing my own seeds and soil amendments.

Hi Joseph,
Thanks for the response. I think, like with most things, it will take an integrated approach and vigilance to make a difference. I do feel like I have to be on it though, as the city/county may make the decision to come in and nuke it with chemicals if the whitetop population persists, which would be detrimental to the site overall. The whitetop that is present in the guilds is currently pretty easy to pull, though I know the roots left broken off in the soil will resprout. I just need to get a better groundcover in there and keep hitting the whitetop; exhaust the resources and, like you said, smother the roots.
Thanks again.