Exterior entry door repair
I agree, it sounds like a veneer that has aged over the years, dried up and is delaminating. One of the problems I see trying to repair the damaged areas with a filler, is that other areas may soon suffer the same problem. With that said, if you want to attempt a fix, you could remove all the loose material, sand off any remaining glue etc , then fill the voids. You could use a plastic wood filler, a thinset type of material or even a setting type drywall compound. Fix it muck like fixing a hole in drywall. spread the filler on with a wide knife, and sand it smooth after it dries.  Be sure to prime and paint the entire surface before allowing it to come in contact with moisture or the outdoor elements.  This may be a semi long term fix, but think you will continue to see problems in the future. Consider a new door soon.