People have used these plastic/geotextile things a lot here. They were very convenient to use as you can buy them with the holes already made to put in the seeds / plants.
We've done the cardboard thing a few times. If you really want to kill the weeds you need cardboard so thick that it won't compost before a few years. It's still better in the sense that it's much easier to recycle.
For us now what makes sense is either planting a cover crop, using hay (but you wouldn't have that at the allotment) or leaving what was planted in place until early spring, which make older people here go crazy as they like to see a clean bare soil for winter

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