Winter, during asparagus dormancy, would be the best time for relocation of the roots. You will have to delay harvest one year after relocation. Mulch beds with a high-nitrogen compost each spring before spears emerge and again in the fall. A light straw mulch will provide winter protection and help to keep weeds down during the growing season.
Glyphosate herbicide (nonselective) is labeled for use on asparagus as a soil treatment or a postharvest application to eliminate grass and weeds. Do not spray the asparagus tops (fronds).
Consider making this area a raised bed. Build a 6- to 8-inch-high box around the perimeter of the asparagus bed. During the first year on the new program, hand removal of grass and weeds will be needed; thereafter, control will be maintained by renewing sufficient mulch.