You don't say how old the new kitten is. It would be best to keep it separate from your older cat for a while for two reasons - one being that you don't know where it has been or whether it carries any sicknesses it can pass onto your older cat. The other being, as you have already become aware, there is a risk to the younger kitten from boisterous play.
It would be best to take the kitten to your vet for a good checkover to ensure there are no illnesses that are contributing to its general weakness. But older and wiser heads than mine can give you more detailed advice on its care and maintenance.
From your description it sounds as if your older cat is taking on a fairly maternal role with the kitten (washing it and so forth) and this also includes disciplining it. That can look pretty rough to the uninitiated and can involve pinning the kitten down. But while the kitten is weak it is best to err on the side of caution.