I had a heck of a time figuring out how to feed my kitten when he first came to my house. Previous owners didn't allow him to ween long enough and put him on generic adult cat food (vet guessed at 3 weeks). He was pretty ravenous when he got to my home, and no baby fat considering his age. I got him at 2am, he announced I was his owner by mewling in my garden incessantly. I only had ham in the house to feed him. Minced up fine, he ate a portion as big as his head and wanted more. Less is more, because digestive upsets are something vets will warn you about.
My Kitkat has no tolerance for meowmix. Science diet kitten chow he ate occasionally, but purina kitten chow was only tolerated if I wet it as if it were gravy train. It didn't occur to me that his changes in appetite were about teething phases. Hindsight being 20/20, his cat chow was being vomitted up in whole pieces (he wasn't chewing, but trying to swallow whole). I was scrambling around trying everything, but making everything worse!
You didn't mention if there was excess hair in the vomit. The hairball remedy in my house is a cat treat he gets twice a week. I also tried kitty greens which did seem to help him (and the rest of my plants were attacked less as result lol). I also came to see that feeding him tiny portions spread throughout the day made his behavioral problems relax; he didn't fear starving anymore and didn't have to resort to extraordinary means to sing for his supper.
I remember my ex landlady's cat, much older and prone to allergies, would vomit often. Cuddles responded well when I'd use a drop of peppermint oil diluted with a carrier oil on her coat. She'd lick it off and it calmed her system down. I don't think they ever narrowed down all she was allergic to, but the peppermint helped reduce the frequency & helped her bounce back quicker from upsets. Plus she smelled pretty.