This may not help you now, but I have found that the following prep eliminates any puckering for me.
I starch the backing fabric. Spray starch seems to get away from me, so what I do now is lay out my fabric on the kitchen island and "paint" it with a diluted starch solution I make myself. I use the blue bottle of starch you can buy in laundry aisles, diluted either 1:1 or 1:2 with water, using an actual large paintbrush! I give the fabric a half-hour or so to absorb the starch well, then throw it in the dryer. When I iron the fabric, it becomes quite stiff. This stiffness means the fabric won't have a chance to bunch up when I am machine quilting the sandwich.
The other thing that really helps is spray basting the sandwich together, rather than pinning or hand basting. The spray baste makes a very firm sandwich and holds all the layers tightly together, so the fabric you can't see doesn't really have a chance to pucker.
If I could do only one of the above to prevent puckering, it would be to spray baste the quilt sandwich.
For repairing your current quilt without taking it apart, I would probably try to starch the backing fabric after taking out the stitching. You could use several layers of spray starch to do this. Just be sure after spraying to give the starch a few minutes to sink into the fabric before ironing; otherwise the starch can stick to your iron and/or flake off the fabric. Be careful too, not to have the iron so hot it scorches the starch. Scorched starch will wash out, but it's not a good feeling to see it! I would probably use a steam iron held an inch above the fabric first, to help the starch sink in and also to see if the backing fabric will shrink a little to get rid of the excess in the pucker. I'd finish with regular ironing.
Mary