Yes, you can.
While technically a page open in Chrome browser tab can't directly check whether a home screen shortcut exists, the page's local data (localStorage, IndexedDB) is shared between home screen instance and browser tabs, so this can be used to communicate the existence of the home screen version.
- Detect if the app is running from home screen
- If it's ran from home screen, save this fact to
localStorage
- Profit! (by reading this from
localStorage in any tab)
When the app is in standalone view (which is possible only when launched from home screen), the CSS media query (display-mode: standalone) will match. In Javascript you can read it using:
matchMedia('(display-mode: standalone)').matches
(BTW: the non-standard iOS equivalent of this is navigator.standalone, but iOS doesn't share state between home screen and Safari, so you're out of luck there).
However, instead of custom instructions I suggest meeting Chrome's criteria for "progressive web app" and let Chrome do the prompting for you.