The Appointment’s slots must have an option to make them bookable once time (with a memory) so other people can’t book them another time (have to be ripped off the calendar).
In the email, the clients must receive a link to cancel the appointment and make the slot available again.
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Hi there and welcome to the Community, @user29448 
Our Appointment Booking widget requires integration with Google Calendar to automatically mark booked slots as unavailable:
Here’s a tutorial on how to connect your calendar: Syncing Google Calendar with the Appointment Booking Widget
Please try it out and let me know if it worked.
As for the option that lats users cancel booking, this idea is already on the Wishlist. Feel free to upvote it 
- End user can change or cancel a booking