Pull unwanted sprouts as soon as you see them. A ground cover is unlikely to stop an existing weed issue, but can help keep one from developing, once under control. Once you pull the sprouts of seeds there, not letting any weeds drop new seeds, you should have a LOT fewer sprouts next year. I would pull diligently this year, then consider installing the ground cover next spring. Something short and thick, like Lamium or Veronica might be nice for your zone. Then you'd want to monitor it closely for any unwanted sprouts until it fills in solidly.

As far as mulch goes, wood chips are known to do a terrible job at weed suppression (and soil enrichment, moisture retention.) A finely shredded hardwood would do a much better job of this. Add to wood chips if you want, no need to remove the chips.

You might also get good results this year by scraping the chips away from under the shrub, layering some newspaper, overlapping well so no cracks to exploit, cover back up with the chips. Any new sprouts under there won't be able to grow.