Adders are protected species, so you can't kill them. If anything, you should be extremely happy that you have snakes in your garden, as they're extremely rare in this country.
Chances are, they will leave the cat alone, as cats are extremely versatile. Remember, cats are born predators :P
Also, if you are bitten by the adder, there is a very small chance you will die. It will hurt, but a trip to the hospital will see you fit and well in no time.
Plus side of having snakes - no rodents in garden. And with the cat too, that means you can have a pest free garden.