I have tried keeping the polythene in place and putting leeks in through the same holes cut for the potatoes. The leeks grow huge, partly because of the rich soil, and because they are far apart, but of course you don't get very many. I still had the problem of some weeds coming up through the holes, and of the terriers digging underneath and tearing the fabric.
I'm a great believer in green manures, in principal, but I don't use them. I once tried grazing rye but it grew about 8' tall, and was far too tough, with amazing roots, so couldn't be dug in.
Now I can't find anything which would survive our winters, except maybe clover, but that needs 2 or 3 years to be worthwhile. The weeds outcompeting the green manure is also an impossible problem.
Ideally, I spread a thick layer of partially rotted manure, or hen house cleanings, then cover the lot with a tarp or polythene. By spring the soil underneath is wonderful, and warm too. You can then cut holes and plant things directly without having to lift the plastic or do even a token dig.
Another thing to follow on after new spuds is pot grown winter brassicas. This depends on when your first earlies are lifted - here it's a bit of a rush, because the spuds are ready later than down south, but the brassicas need to go in a bit earlier, ideally. But for you, that could work well. You could even sow a low growing green manure between the brassica plants.