Tenant Wants A Cat

Tenancy Types and Management

I've recently signed a 12month AST with these tenants. They have now asked if they can have a cat. We stipulated no cats in the AST and they were aware of that but have now said the cat will be well-behaved, they'll pay for damage etc etc. However I've had another property wrecked by a cat, coffee table, sofa and skirtings scratched up and a carpet that needed defleaing.
I'd be prepared to let them have a cat but don't want to just take their word for it that it will be cleaned - is there some sort of offical 'addendum' to an AST that I could draw up saying that they must have the carpets professionally cleaned and defleaed at the end of the tenancy at their own cost before getting their deposit back (receipts to be provided) and they'll pay for any scratches to be made good? Is that an acceptable thing to ask?
Many thanks
Emma