To make hasty, careless decisions; to discard something after encountering a setback. The image alludes to throwing away the handle ("helve") of a hatchet after the blade has broken off. Come on, I'm sure we can fix it—don't send the helve after the hatchet. My little brother always gives up at the slightest problem, no matter how many times we remind him not to send the helve after the hatchet.