Another trick, actually a variation of that of @agentp, is to use wire mesh (also called hardware cloth) instead of a wood strip. Choose one with small holes, 1/8" if you can find it, 1/4" is real common in the building stores. It is particularly useful when the thickness of the material is thin (as in the case of interior hollow-core doors 1/8" luan or mahogany plywood is typical)resulting in a patch that would be thin and prone to cracking out. The holes in the mesh allow for more of the filler to pass into it and lock the patch in place, much like wire lath in traditional plaster work. The down side is it is harder to glue in place and needs a more viscous product like liquid nails. The basic procedure is the same.
- Cut a strip of wire mesh that will just fit into the hole width-wise but is several inches longer.
- Apply ample Liquid Nails or similar viscous/thick general purpose construction adhesive to the ends of the mesh strip.
- Tie string on the mesh such that they will be near the edges of the hole.
- Thread the mesh into the hole, holding the strings so not to lose
the strip in the door cavity.
- Tie the strings to a wood slat that spans the hole, paint stirring
sticks work great and can be picked up for free.
- Give the construction adhesive plenty of time to dry.
- Patch the hole with a quality wood filler. Plan on doing it in a
couple coats since the fist coat will want to press though the mesh
and be nearly impossible to get smooth.
- Sand, prime and paint.
Much more involved way but potentially more sturdy repair.
Another way to go about fixing holes in paint-grade thin panels is to fill them with with a plug. But it assumes that you have a number of tools at your disposal and it if you don't it would probably be cheaper to pick up a new door blank and just replace the entire door. This suggesting may not work on yours if the wood between the holes is too damaged. In the pictures, it looks like it could be split.
You need to have sharp holes saws, and not the one or two tooth varieties for fast wood boring, they will tear thin plywood. You need to have a size that is just a little larger than the hole and then another that will cut a piece that is just barely under the size of the hole cut by the other hole saw. Or you could use the same and just use more filler to make up the 1/8" or so difference. You will need plywood the same thickness as the door panel, typically 1/8". You will need at least one piece of plywood 1/2" to 3/4" thick and nearly as wide as the door that will become a hole-saw guide. You will need two scraps of plywood several inches square and two clamps that will open wide enough to accept the hole-saw guide thickness, the door thickness and the scraps thickness. Polyurethane wood glue, like gorilla glue for example. Wood filler. Minwax make a "High performance" wood filler that is a two-part system that works well and is very similar to Bondo, just not red. Putty knife, sanding block (can wrap sandpaper around block of wood is fine), sandpaper, window scraper, and wood slats like paint stir sticks.
Hole saws are guided by a pilot bit, but that means there has to be material for the pilot bit to bore into. You take the piece of plywood that will be the hole saw guide and set it across the door width next to a hole and mark on it were the hole is and then position it carefully so the mark is the center of the hole, clamps it to the door using the scraps to protect the other side of the door. You then use the hole saw to drill a hole through the guide plywood and into the door panel, do not drill into the opposite door panel with the pilot bit or hole saw. This takes careful control and the door should be taken down and put on saw horses by the way. Also, the pilot bit can be adjusted so that only a 1/2" or so is past the hole saw. You do this for all the holes in the door.
Then you use glue the slats into the holes as you would in @agentp advice above. While that glue is drying, cut enough plugs from the 1/8" plywood with the larger hole saw. Use a scrap of plywood as back and clamp them together and onto a study work surface. Use some sandpaper to clean up the splinted edges and make sure the back is flat. Glue the plugs to the wooden slats in the door. Spread the glue thin so you don't have the plug higher than the surface of the door. Place a weight in the center of each plug and let it all dry.
The polyurethane glue will expand out around the plug. When it is dry you can slice it off with a sharp window scraper, the kind with a single edge razor blade in it. The mix the filler in small batches very well but not too long (follow the instructions on it) and fill the hole in the center of the plug and the around its edges. Work quickly and don't try to go back over the same place too much. Better to have to do more coats than deal with a mess from dragging the patch that is curing out. Leave as little patch on the surface of the door as you can. More careful coats are better than few sloppy coats. Clean your putty knife with the window scraper and some acetone or MEK in between mixing batches. Also clean the mixing container well too. Partially cured patch will cause new batches to cure too fast.
Let the patch dry until all the tackiness is gone and then sand. 100 grit, 150 grit then 220 grit should give a nice smooth surface. You will have to smooth into the existing paint films when sanding to make the patch invisible.
This will be a lot of work but will make a better patch. When done very carefully it can even work with non-paint grade doors and wood paneling patches but can be hard to match grain and pattern and is never really invisible, in which case you would cut the plugs on a drill press with a circle cutter and use a forstner bit on the door panel if possible (depends on size of hole). But with that many holes that close together, it may really be a new door blank is the better way to go. If you have a Habitat for Humanity ReStore in your area you may be able to get a door blank or even a pre-hung and just give them back the frame, for less or equal that it would be to buy supplies and maybe tools if you don't have them to try and patch the door.