Odd place to put weight on a road bike, probably indicating a fit adjustment is needed. You might want to raise the bars and/or bring them back to reduce the weight on your hands until you can ride comfortably for longer distances. If they are a good height, you will find yourself leaning lower and sliding your hands down the outside of the bars frequently, hooking more with your thumbs while putting out stronger efforts. Lower them when you spend most of your time in the drops without pain.