When I download apps they go straight to the App Library, this is the only place I can access them from (when I go to an empty slot on my Home Screen it doesn’t appear)

Settings > Home Screen > Add to Home Screen [Tick]

Probably you have selected [App Library Only] for Newly downloaded apps. See the pic below

You can also drag the apps from the App Library to the home screen, as described here:

Open apps on iPhone - Apple Support

• Add an app to the Home Screen: Touch and hold the app to open a quick actions menu, then choose Add to Home Screen (available only if the app isn’t already on the Home Screen). The app still appears in the App Library.

The option to add new apps only to the AppLibrary will automatically be turned on, if you have hidden a page of your home screen.

• Note: When Home Screen pages are hidden, new apps you download from the App Store may be added to the App Library instead of the Home Screen.

In the App Library press and hold on the app until it begins to shake, then drag...