I would try squirting a small amount of wood glue into the crack, then using blue tape to hold the piece in place until it dries completely. Use a bit of spackle to smooth out the edges after it has dried, sand and then touch up the paint. Probably won't be perfect, but should look pretty good.
Wood glue and wood filler are my go to for problems like this. There are paintable wood fillers and glue that dries clear or white. A good sanding sponge and you should be good to go. Do you know what caused it? Maybe something is catching on it that could be minimized.
Minwax makes a wood epoxy just like wood filler and you mix it and put in the crack so it doesn't expand or crack anymore. Caulk will crack and so will plain wood putty.
I have fixed that same area. I used Bondo wood filler, but you could use Elmers also. I taped the area off and then filled in the cracks. I let dry completely, then sanded the area down. Mine needed 2 applications. When dry, you can prime and paint.
Looks like you have maybe tried to use glue of some kind already? If that is the case, you must let it dry thoroughly and then plane it down flush with rest of door frame, before priming, undercoating and then painting.
If you'd rather replace the piece it's a standard trim piece size from any big box store.
I have fixed that same area. I used Bondo wood filler, but you could use Elmers also. I taped the area off and then filled in the cracks. I let dry completely, then sanded the area down. Mine needed 2 applications. When dry, you can prime and paint.
I have and it is much easier than repairing the whole, wood, I use, glue, and paper, once it harden, i sanded and then painted.
Looks like you have maybe tried to use glue of some kind already? If that is the case, you must let it dry thoroughly and then plane it down flush with rest of door frame, before priming, undercoating and then painting.