Can you make a separate iCloud email address?

Depends on what you are looking for exactly.

As mentioned Hide My email can generate new email addresses when signing up to websites that support the option. But only during sign up.

If you mean actual functional email addresses you can manage yourself and name whatever you want, you can create up to 3 using the iCloud email alias function at https://icloud.com

click here ➜ What are email aliases in iCloud Mail? - Apple Support

You can make a nearly infinite number of aliases, if that helps. The "Hide My Email" feature creates random "burner" email addresses, all of which get forwarded to your real iCloud address, so they all come to the same mailbox. Not sure if that's what you're looking for.