Covering soil with something like the impervious plastic limits the necessary exchange of air that good, healthy soil needs, so previously healthy soil now becomes anaerobic instead of aerobic and that is bad. A better material to use is newspaper and shredded leaves, does the same thing as the plastic (denies the plants access to light) but allows both air and water into the soil and adds needed organic matter too, something that plastic will not.

stirrup hoe can be used in areas where the plants like vegetables are grown in a line. It has a hoop in the shape of a square or circle. The hoe cuts the seedlings and can also uproot them. This helps in ensuring that the weeds do not emerge again. But in places where the weeds are growing quite close to the main plants, you need to remove them by hand as there is a chance of damage by the hoe.

Mulch is also effective in reducing weed growth. It works by preventing the sunlight from reaching the soil thus inhibiting any growth of weeds. Clippings of grass, branches, hay and other degradable substances like leaves and barks can be used.

Here is a link that might be useful: Organic Weed Control