kitten bullying my older cat

Answer 1

Your cat could be under more stress from the new cat being there. Is this new cat a permanent fixture I had keep the cats nails trimmed and if all else fails have the cat declawed. Some people think this is inhumane but if it finding the cat another home is not an option, I had go for declawing.
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Answer 2

there not bullying your kitten is just playing ...... let them... it's FUNNY!  MY KITTEN DOES THAT TOO MY DOGS ha, ha. he scratches my dogs.