Hello Michelle, I think you need to do more weeding of your asparagus bed in summer and autumn. Make a habit of little and often, every week or two to keep it always clean. It should be a quick job. then a light dressing of compost in the autumn.
However from the weedy state you describe, I would keep the polythene on (yes its a good idea) until mid April if its cool weather, or early April if warm for the time of year. Maybe lift in late March to check there are no spears pushing up, remove if so.
In case other readers may wonder, non-weedy asparagus beds do not need polythene mulch.
