My cabinets are pressed wood. can they be repaired

We're looking at a door here, right? If you use a razor blade or razor knife to cut the plastic laminate off at a parting line then sand the wood and remaining laminate you could paint this. I'm assuming that this is not urine damage. If it is urine damage you need to replace the door. You will not get the odor out and the paint may not adhere.

Step by step guide on how to paint pressed wood.

https://www.hunker.com/13412945/how-to-paint-pressed-wood-furniture

On the cheap, try covering them with wood grained contact shelf liner.

WOW! You can remove the fake paper wood grain. Patch any damage with wood putty. Sand them smooth and paint.

Make sure they are clean and dry. Remove the doors and hardware. Mark the doors and cabinets with tape where they go. Lightly sand the doors and cabinets to remove any gloss and roughen the surface for paint with 120- to 220-grit sandpaper. Use a tack cloth or damp rag to remove dust after sanding. Prime with a stain blocking primer like Zinsser 123, KILZ,or BIN and have it tinted to the color of the top coat. This will prevent dark or stained surfaces from showing through the top coat. Acrylic, or water-base, paints are low-fume and clean up easily with water. Alkyd, or oil-base, paints require good ventilation because the paint contains solvents that can irritate your lungs and make you feel sick. Alkyd options require mineral spirits for cleanup, but they provide a hard, durable paint finish. Whichever you use, buy the best-quality paint you can afford for a lasting kitchen cabinet finish. Seal with at least three coats with a water based polyurethane. Use a small foam roller and foam brush for a smooth finish.

This guy has some good tips on his video. It always helps me when I see how things are done, hope it helps you.

https://www.youtube.com/watch?v=2_nfB27o3WQ

Don't waste your money and time trying to cover them with contact paper.

The surface is so disintegrated and uneven that the paper will not adhere adequately. Any stickiness that does linger on the cabinets will attract dirt and lint like a magnet, and you will be faced with trying to remove a gunky mess.

If it was my project I would sand off the damaged areas and use poly to fill In uneven areas. Clean and sand the rest, prime, and paint with something durable. Maybe change the hard ware as well or if you’re broke, paint those too.

Remove the damaged pieces, sand, prime and paint.