The stronger your core, the less weight you need to put on your hands. If you think of a triangle with hands, butt, and shoulders as the apices, the longer your hand-butt line is, the less weight on your hands.
But is it the weight that is causing fatigue? Where do you place your hands, what parts of your hands touch the bars, and are your elbows locked or bent? Bent elbows reduce stress on your hands. If you place your weight on one of the nerves, you stress you hands. Someplace in BF there's a really good write-up of cycling and hands, but I've lost track of it. With a little luck, somebody will be along with the link.