Give your garden a good soaking. This also helps keep the paper and grass from flying away and gives your plants a good drink. The space you left around the base of the plants allows plenty of water to get through immediately while the grass and newspaper will allow the water to soak through more slowly.
You may need to do some minor weeding around the base of the plants but for the most part 95% or more of the weeding you normally do will be eliminated. Add more clipping during the year as needed. At the end of the year, rototill the clippings and the paper into the garden. No need to pull anything up at all. I find this works many times better than just clippings alone.