Cut the bad cork out using tape rings wrapped on the grip as a guide. Replace with new cork rings cut to that length. Groove cut the interior with a coping saw to about 1/2 the cork's thickness, break the cork ring, the ragged edges hide better than clean edges when reglued. put together with tape and ream to blank size than glue in place. Sand to the shape of the grip.
Fastest and easiest way is to buy a piece of the graphite grip Lamiglas sells and reduce the cork grip to the ID of the graphite tube, that slip over the grip and glue on, install new end cap and your done.
Forgot to mention another easy fix, plastic grip tubing. Comes smooth or X-Flossed acts just like electrical shrink tubing, it doesn't lose length from what little I've used it, so you repair the area with cork pieces, wood putty, anything that can be reshaped back to the origin grip than cover the grip with the tubing, heat with a heat gun or high temp hair dryer to shrink it down to the grip's shape and dimensions, trim if you need to a bit than add a new butt cap and you have a good loking and very durable rear grip.