Yes. You can refuse a specific holiday request, as long as you are not preventing the employee from taking their statutory minimum leave across the year. The usual approach is to give notice at least as long as the leave requested, for example two weeks' notice to refuse a two-week request, and to apply your approach consistently.
Refusing leave fairly is about genuine business reasons and consistency, not picking and choosing. Blanket or inconsistent refusals, especially where a protected reason might be involved, are where complaints tend to start.
Every situation has its own details, and those details change the right answer. For a read on yours, take the free Employee Situation Check or book a call.
General guidance to help you decide your next step, not legal advice. Employment law changes, so always check the current position for your own situation.