Can I add a different Apple Id

You can sign in with a different ID but you cannot have two IDs on an Apple mobile device (exceptions: dual SIM iPhones, but we don't need to go there, and some signing into select services with a second ID). Because of the exceptions it would help to know you objective for a different ID. Incidentally, you have to sign out of the first ID before signing in to the second..

Note that changing IDs can lead to mixing of content so it is best only done if you are willing to erase content in-between, or if you have a particular reason to mix content.