A crop or whip is used to reinforce leg aids if the horse isn't responsive. Many people follow a three-step rule, first asking with a squeeze, following up immediately with a sharper kick if the horse does not respond, and then backing that up with a squeeze + tap of the crop if the horse does not respond to the kick.
Also, I have been told on occasion to use a crop to remind a horse not to bulge through his shoulders. I was leg yielding with a dressage whip, working on getting proper leg yields without him bulging out. At one point the horse stopped listening when I would straighten us up, and then try again; he just kept going out through that shoulder. So I rode with my dressage whip before my leg and was able to give a gentle tap to his shoulder, just to remind him to keep it under control. After that he began listening to my usual aids once more.
If your horse is responsive to your leg you won't need a crop, but you may like to carry one if the horse is difficult to get moving. I know lots of the lesson horses at my stables will be much more forward and willing if you simply carry a crop without ever using it!
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