Hopping I find usually describes a hopping in the rear brought on by excessive rear grip followed by a quick release of grip and then the tire re-gripping. This causes the kart to hop as the inside rear lifts and sets down repeatly during a corner. Its fairly excessive and easily felt from the seat. Hopping can be a handling/tuning issue, or can be brought about by over driving the kart in a corner. Tuning-wise, its usually brought about by too narrow of a rear width, center of gravity to high, or rear axle too stiff.
Push-Kick is a severe handling issue where the kart experiences push going into a corner followed by an immediate loose condition when transitioning from brake to throttle. Its usually indicative of a balance issue (too much weight to the rear) or an overall lack of mechanical grip.