Here’s a good way to draw a damaged wood plug out of its counterbore, provided the glue is still wet. Drive an auger-tip wood screw into the center of the bad plug. When the tip of the extraction screw bottoms out on the head of the seated screw, keep turning it slowly and the threads will lift the plug out. Be sure to use an auger tip screw for this job — it will bore a pilot hole in the plug instead of expanding it and damaging the counter sunk hole.
– Mark King