I don't do the covering in plastic thing to clear weeds . I dig up as much of them as i can , including all roots . Then once clear i allow the weed seeds to germinate . Clear that flush of weeds when they are 1-2 weeks old . Rake and level , allow the weed seeds to germinate again , clear as before . Each time you do this the amount of weeds that pop up will get much less , as there will be less weed seeds in the soil . After 4 or 5 flushes there will be hardly any weed seeds and any root that escaped the fork will have popped up and been pulled out , result , a clean veg bed that requires very little weeding .
Leave any manure for a week or so to flush the weed seed also .
I would avoid the communal compost bin too .
I avoid newspaper and cardboard now , due to chemicals in paper/cardboard and print . I do burn it and use the ashes though , i think most of the chems are gone then ?
You can pop in broad beans , round seeded peas , cabbages maybe ?
But clearing would save lots of weeding later on .
Mind you , not many weeds will germinate now lol .