I have tried few things equivalent to turnips, such as Tagetes minute against couch grass, and have not been impressed with results.
Tris’ plan is best IMO. Two ways to mitigate loss of space:
1 add extra compost to a small, intensive cropped area where you can grow many veg that grow at close spacing, and be super thorough on weeds
2 Plant a few squash, courgette and any wide spaced veg through holes in the poly-covered area so its cropped a bit. Be sure to pull any weeds popping up through the holes.
