Wipe around the inside of the cabinet hole with fine sandpaper and a dry rag to remove loose material and dust from the hole, so the repair isn't bonding with loosened wood shavings.
Fill small holes with an epoxy repair putty. Mix the epoxy in accordance with instructions on the package to ensure the best possible bond. Wipe away excess with the rag and allow it to dry.
Trace the size of larger holes onto a piece of paper with a pencil, then draw the hole on another piece of particle board. Cut out a replacement piece of particle board with a saw.
Fill the inside perimeter of the hole with wood glue, then apply wood glue around the edge of the replacement piece of board.
Slide the replacement piece into the hole, matching it up carefully so it slides in smoothly. Wipe away any excess glue with the rag.
Push a flat wooden block onto each side of the hole, then clamp in place using a vice or C-clamp.The blocks hold the hole repair flat and ensure it doesn't move as the wood glue dries. Let the repair dry overnight.
Remove the blocks and apply paint to the repair if necessary with a brush to blend the repair in with the rest of the cabinet around it.