How do I deal with this???

This is an unoccupied area in my garden where I’ve put down cardboard and several inches of wood chips. The fricken dollar weed is over taking the wood chips. I’ll never be able to plant in this if I can’t get rid of the dollar weeds. Do I have to rake all these out by hand? Cover it with tarp? I don’t really want to disturb the chips too much. And I don’t want a dollar weed lawn.