I can answer this coz I have recently done the same thing.
You're right in thinking this is a good idea!
You use the add-on domain feature in your cpanel. Make a new add on domain for mydomain-acme.co.uk, with the name 'acme'. This will create that directory 'acme' under your public_html folder. All you then need to do is change your mydomain-acme.co.uk dns settings to point to your eukhost dns, you don't need to transfer the domain. You will get separate mailboxes for your new domain.
I have a few
website hostings, so I set up a new account with the 'root' account being a web design business, then all my other
website hostings being addon domains. Makes like much simpler!
Chris