Cabinet doors are typically made in one of two styles: a solid or flat door, or a panel-style door. With a panel-style wood cabinet door the panel is often a loose insert that is built into the cabinet door frame during the manufacturing process. This means there is a groove running around the inside of each side frame and the panel is slotted into that groove. If your panel is damaged you have several options, depending on the extent of the damage.
Remove the cabinet door from the cabinet. Wash the door with soap and water to remove as much grease and dirt as possible. Dry the door with a towel.
Examine the damage. If a hole was made through the panel it is likely the wood is torn or missing. If you can glue the wood back together that may be worth a try. Use carpenter's glue between the wood pieces. Place scrap wood on the top and bottom over the broken area and clamp the scraps together to place pressure on the wood pieces while the glue dries. After the wood dries use wood filler to fill in any holes.
Remove extensive damage that cannot be compressed. Clean up both sides of the panel with a utility knife or small saw until you are left with good wood. Mix an epoxy filler used for furniture repair. This will knead into a dough-like consistency. Press the filler into the gap from the back side of the door. Smooth it out with a putty knife. Place a scrap piece of wood with waxed paper over it under the repair and turn the door face up. This keeps the filler from pressing back through. Smooth out the surface filler. Leave it slightly humped to allow for shrinkage. Let the filler dry as suggested by the manufacturer. After the filler dries use a regular wood filler to fill in any low areas.
Sand the dried repairs on both sides of the door. Use a sanding block and sand in the direction of the wood grain. Sand the repair until it is smooth and flush with the rest of the panel. Wipe with a tack cloth.
Stain small repair areas to match existing stain. Paint larger areas using acrylic paints. Thin down the acrylics with water and blend colours to exactly match the wood tone. Fade in a base colour to match the background wood and then lightly use a tiny brush or feather to simulate a few matching grains so that the wood filler looks like the original wood. Allow the acrylics to dry completely and then use a sealer over the entire panel. This should make the repair nearly invisible.
Router out the back inside frame edge if the panel is too damaged to save. Cut a piece of glass to fit. Run a bead of silicone along the inside front edge of the frame and press the glass into the silicone to hold it. You can insert glaziers points for additional security. This will turn the panel into a glass door.
If you need to replace the panel with glass, consider doing a companion door if the cabinet uses two doors. This will make the repair look like an intentional upgrade. Repaired panels can also be painted rather than stained if you want to paint all of the cabinets to hide the repair.