Is there any way to fix the broken corner of a particle board drawer front?

Nails are good for some things, but they're not good when it comes to resisting withdrawal, especially from end grain. That's the force you applied to the nails when you pulled on the drawer front, so I'm not surprised they pulled out.What you need instead is a good woodworking glue. I'd recommend a polyurethane type because it's so strong. The nails simply hold the pieces in place until the glue cures.Make sure the edges of your drawer frame and the area behind the drawer face are clean and smooth. Wear gloves (to avoid staining your skin) and apply a small amount of glue to one surface. Attach the face to the frame with a few finish nails. If you have them, clamps will help hold the drawer front while you hammer away. Sink the nail heads below the surface with a nail set, and fill the nail holes with wood filler before you touch up the finish. Let the glue foam and expand as it cures. When the excess is dry, cut it away with a utility knife or chisel