Despondently tells you how someone does something: in a way that shows they are extremely unhappy and have almost given up hope. It usually describes a look, a voice, or a small action — shaking your head, staring at the floor, putting a letter away — rather than a big, noisy scene. The word is a little formal and bookish; in casual talk people more often say sadly or in a really down way. Use it when the sadness has that “what’s the point?” feeling inside it, not just when someone is briefly upset.