I would point out that people often misuse the cavesson. Really the cavesson is supposed to be nothing more than decoration, but people often use it to keep the horse's mouth closed. To do this the cavesson must be over-tightened, and then it crams the horse's cheeks into his molars, which is not a comfortable thing for him. If it's necessary to keep the horse's mouth closed one should use a dropped noseband or a flash cavesson.
Some people also believe that the cavesson holds the blinkers in against the horse's cheeks. While a "noseband" that is nothing more than a strap that runs through the cheeks & buckles around the nose, it may serve this purpose somewhat. The thing is, your higher end bridles have a separate cavesson which does not run through the cheekpieces, and this proper cavesson will do absolutely nothing to hold the blinkers in against the horse's cheeks. A well made, properly fitted bridle should not gap away from the face, but if it does to some extent then a shoelace run from cheek to cheek, across the nose, will fix the problem. A narrow little lace will be barely noticeable, and won't squeeze the horse's cheeks into his teeth.
In harness a martingale isn't so much for the driver's hands raising too high--after all, with the lines run through the turrets no matter where the driver's hands are, the lines always run straight from turret to bit! When riding, yes, the martingale can compensate for the rider's hands being too high. In both riding and driving, if the horse sticks his nose up & out, without a martingale a snaffle bit will pull up into his molars--the horse can evade the bit this way, and at the same time may resist even more because the bit--especially a jointed snaffle--will crush his cheeks against his molars--again, causing him pain, and this in turn creates more resistance. If the horse is wearing a martingale & shoots his nose up and out, the martingale will cause the reins to continue to work on the tongue and bars--the downward pressure created by the martingale prevents the bit from riding up in the mouth. Thus the rider/driver has better control of the horse.