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If your digging the weeds out then the roots will be dug out with then,You'll never get them all out but weed killer is a no no especially if your planting straight away,As GJ says lay down cardboard and plant through it,I'm afraid there is no quick fix, But as I've found out it's hard work running an allotment and many fall by the wayside because they don't realise the time and effort that goes into running a lottie, Once the hard work of digging it, is over, the work will be easier but you have to keep on top of itHoeing the weeds should be done at least once a week Best of luck and happy growing.