Apple Mail & Gmail

Hi there, I have recently created a Gmail (Workspace) account for my business. I have used Gmail before , but I did not have it linked into my Apple Mail.

My problem is that if I get an incoming email in Gmail and I move it to a folder in Gmail , it still remains either in 'Archive' OR 'Important' folder in Google in my Apple Mail.

Not wanting to double up on space , when I delete the email in my Apple Mail it automatically deletes from my Gmail folder.

Is this normal? Can this action be fixed & if so what are the settings for me to do so?

Thank you.