My cat has done this all of his 14 years. They eat a bite of food and then 'chew' for a while after, a few seconds, almost like they have something stuck in their teeth. He doesn't do it all the time, you can hear it, it sounds like he's chewing on a rubber band. Normal? I don't know, but since it's not all the time and he's done it for so long it's just became a part of him.
Cats aren't ruminates (animals with four stomachs such as cows) so they can't "chew the cud"
Do you mean your cat is throwing food up and eating it again? Or maybe your cat is grinding his / her teeth and/or smacking the lips? If your cat having difficulty eating or swallowing food? That could indicate a dental problem.