How to repair peeling on cabinets

I will echo the Bondo route that others suggested. The big box hardware stores sell quart cans of general purpose Bondo in the paint aisle.

Cut out the vaneer over the bad area with a razor blade or box cutter.

Sand with 80 or 120 grit to remove any loose bits of the board. The particle board will crumble a little, but it will be fine. Just dont go overboard. Vacuum it to remove most dust.

Using a flexible putty knife, get one to two globs of Bondo on a clean piece of cardboard. Don't do too much as you will have about 5 minutes of working time once the hardener is added. Follow the instructions on how much hardener to add, you wont need much. Fold the hardener into the Bondo with the putty knife until it is a uniform color.

Spread the bondo over the area you cut out. Try to get it smooth, but dont overwork it. It doesn't have to be perfect.

Using 60 or 80 grit paper, sand it as it starts to dry. If it is gumming up the paper, it is still too wet to sand, give it another minute or two and try again. You are shaping it with this step. If you let it completely dry, it is hard as a rock and more difficult to sand.

Once it is in a good shape, switch to the next step up for sanding and repeat (60 to 80 to 120 to 180 to 220). Do not skip gits or you will just make work for yourself. You can probably stop at 180, but since it is next to vaneer, i'd go to 220. Again, if it is gumming the paper, wait a few minutes. Once you are at 220, use it to sand the entire cabinet to rough up the surface and accept primer better.

Once sanded, wipe the entire cabinet down with a tac cloth. I also recommend wiping the cabinets with a wax and greese remover so everything will stick better.

Now it is time for primer and paint. For a smooth finish, use a good quality foam mini-roller and foam brush for tighter areas.

Prime the cabinet using any water-based primer; killz is pretty good. They make an oil based that would be better, but will smell up your whole house, so I don't recommend it unless you have excellent ventilation.

Lightly sand the peimer once it is fully dried with 220. If any spot is too light, do a second coat. Wipe up the dust.

Paint with an alkyd paint. Do at least 2 coats.

Not necessary, but if you want extra protection, you could use a foam brush (not a roller here) and do 3 to 4 coats of a water-based polyurethane on top of the paint. It might be overkill for what you want though.

This might seem like a lot, but you will have a great finish when done. Also, fair warning, Bondo has a strong smell. Open a window.