Hi Kate,
I had similar situation with new ground I took on about two years ago, though not Hoary Cress ( like Charles I have never heard of that) just lots and lots of perennial weeds. I covered the beds with about 4″ of Horse Manure and then covered them with DPC. It worked brilliantly and when I removed the DPC after 12 months I had the finest tilth I could wish for.
I think of this method as an in situ compost bin.
Don.