The corner with the crack and remaining hole for the pin can be done simply White or carpenter's glue and a wooden peg to fill the hole, wipe off excess glue with a damp cloth, drill a new hole and use a brad with the head cut off for the pin.
The second one I'd remove the broken section. Basically cut a notch from the back, at the step until you reach the break, then cut downward from the edge until you meet the first cut. That give a simple rectangle to fill. That can be done with a short piece of strip wood.
Judging by the color, mahogany.