re-doing resurfaced kitchen cabinets

Several years ago we had our kitchen cabinets "resurfaced" We had no idea that meant putting some kind of substance (type of paper I think) over the top of the original cabinets. Now it is peeling and we don't know what we can do with it. Can't afford new cabinets so has to be something we can do ourselves. We are pretty good DIY'ers.
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