To deal with properly we would need all the usual details, especially the age of the dog. Because if a lurcher (most dogs) wants to kill a cat, the cat's dead. Pinning down, while not so good for the cat, sounds like puppy treating cat like another puppy. Any dog can kill a cat.

Can Lurchers live with cats?

Lurcher SOS hounds that have been tested in a home with cats are termed as cat trainable and are deemed suitable to live in a home with cats, provided some precautions are taken by the new owner at the start of the adoption until the dog and cat know each other (eg muzzling, just in case).

Are Lurchers dogs good with cats?

Being a sighthound means that these dogs are not always suitable to live with a cat, but again this depends entirely on the dog as we have rehomed some that are perfectly happy with a feline housemate.