This is not desired, so the new seedlings should be cultivated and added to the compost pile before they have a chance to flower and go to seed. Incidentally, chickweed or most any other weed that grows in the garden over-winter can also be considered a cover crop, when spaded or tilled into the soil. Remember these weeds must be tilled or spaded before they flower and go to seed again, or you will fight the germination of the new seed all spring and summer. These nuisance weeds are not recommended as cover crops, but since they sometimes appear in the garden why not take advantage of their value as green manure. In addition to cover crops there are other ways of adding organic matter to the soil, over winter.
Animal manures are a great source of organic-humus and nutrients.
They are best added to the soil in the fall or early winter, so the nutrients have a chance to breakdown over winter. Compost is another great way of adding organic-humus. And, when you have finished harvesting your vegetables, simply spade or till into the soil the portions of the plants that still remain, as they also become organic humus, adding nutrients and organic matter. Of course, use good common sense, if a crop has had insect or disease problems it would be best to grub it out, rather than take a chance of spreading the problem in the soil. Fall is a great time to improve the soil in your vegetable garden, and in other parts of the garden too.