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I would thing the most effective option, though not the easiest, would be to buy some thin heat-resistant fabric (lots of options out there, anything that could be used on an oven mitt will do) and stitch it on the backside.
You could use upholstery glue to stick it to the foam for better form-fitting, but you should still stitch it. The fabric would both protect the foam from abrasion and hold the flaking pieces in place. For any parts that already started coming off, I hear water-based polyurethane can help. It must be water-based, other solvents can react with foam.
You'd need to stitch near/around the edges and around the holes for the clips, and you'd want to stitch all the way through the top carpet so it's properly sandwiched between 2 pieces of fabric instead of having the thread cut through the foam like cheese.
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