Voice cues can absolutely be used while riding, and in many disciplines and training traditions they are a standard part of the communication toolkit. The historical resistance to voice cues in some English show ring contexts — where they are penalized in dressage competition, for example — has sometimes given the impression that voice cues are unsophisticated or useful only for beginners, which is precisely the opposite of the truth. The most effective trainers in virtually every discipline use their voice deliberately and skillfully as part of their overall communication system.
For young horses in early under-saddle work, voice cues that were established on the lunge carry over to the ridden context and give the trainer an additional communication channel during the period when leg and rein aids are still being introduced. A horse that already knows trot on from lunging will often trot from a voice cue under saddle before it has fully learned to respond to the leg aid, which allows the rider to get the movement and reward it before the leg aid needs to become more demanding.
For trail riding and general horsemanship, voice cues are practical safety tools that do not require the horse to be in a specific frame or collection to execute. A trail horse that responds to whoa from its rider's voice — even when the rider has dropped the reins to open a gate, or when their hands are occupied — is a safer horse than one that only stops from rein pressure.
In Western performance, voice cues including the cluck are widely used and accepted in training and in some competition contexts. Al Dunning and other top Western trainers routinely use voice as part of their training communication, and the cluck in particular is used across virtually all Western disciplines as a light forward-energy cue that most horses learn to respond to very early in their training.