I rather stupidly removed an app on iTunes Connect, thinking I could re-create it with the same name.
So, when creating the app in iTunes Connect, on the first screen, can I call the app: “Xxxxxx for iPhone”
.. and this will be the name in the store (?). Then could I make the app just show up as Xxxxxx on the actual device when people install it?
Yes, if you type "XXXX for iPhone" in iTunes Connect, that is the name of your app that will show up in the App Store. While il Malvagio Dottor Prosciutto is mostly correct about not being able to use iPhone or iPad in your app name, Apple does allow you to use it if your app name ends with "for iPhone" or "for iPad". That is fine while "iPad Drawing" or anything of the likes is not good.
The name that appears under your app on the home screen of iOS devices is called the bundle display name. This does not have to be exactly the same as the iTunes Connect name, and you can set this up and change it when making your app (i.e. it is not linked to iTunes Connect, it is changeable). Apple requires it to be similar to the app name on iTunes Connect as obviously the users won’t know what the app is if it has a different name.