I’d suggest going to a hardware or home supply store and look for a small bottle of Gorilla Glue. It’s fantastic for fixing wood! You have to dampen the wood before you put the glue in the crack because water activates the glue. Squirt a small amount of water into the crack with whatever you can find; a squirt gun, a turkey baster, something like that. Put some Gorilla Glue in soda bottle cap and force it as deeply into the crack as you can using toothpicks or a small nail. Then find a way to close the crack in the wood. Tape it down tightly, or drive a nail in next to the wood and bend it over to push the crack together. The glue takes a little while to cure, but it’s stronger than the wood itself when it’s done. The glue foams up and expands to seal the crack. Any excess can be cut off with a sharp knife.
You will have to redrill the mounting holes afterwards, and drill them a little deeper this time, but it should fix your problem.