"To peck" is when a bird uses its beak to stab forward to either attack something or eat a piece of food. You add the preposition "at" just to mark what is being pecked (i.e: The geese pecked at the bread). "Pecking around" is when the bird is just pecking at different areas of the ground hoping to find something to eat, like you see pigeons do when people are feeding them at the park. Finally, if the hen is going to "peck at" the worm then it means it is going to strike at it with its beak, normally eating it. Just note that pecking doesn't always signify eating, as it can be an attack too. "To chew" and "chew on" is basically the same difference as "peck" and "peck at". "At" is simply the preposition to show what is being chewed and isn't always needed (like in your example. Both sentences mean the same thing).

I think that both 'at' and 'on' in your examples give an approximateness to the action. Chewing something is what you do when you want to eat it, or you are doing it deliberately and repetitively. Chewing at something is a bit more 'recreational'. With pecking, the action is already a bit approximate, so if the action is aimed at something specific (like the seeds) - it is giving the idea that it isn't very effective. A peck is a single, stabbing motion, so I don't know if it really describes eating a worm. From memory a bird will peck at a worm a few times to stun it, or kill it, then it will pick it up and eat it.