Taking the scenario you have mentioned into consideration where you have deleted an item from the menu and a user is trying to access that item,
Or for that matter in any scenario, error message like the ones displaying in the screenshot you have shared should never get displayed in the front-end (user side).
The error message is giving away critical information which can be exploited by malicious users. Although such details can be of good help to developers for debugging and fixing problems, users don't need to see it.
Instead of allowing default PHP warning and error messages in the front-end a custom message should be displayed like the ones you have suggested in your comment.
Redirect a user to a custom 404 Error page (Page not found) as per the scenario you have mentioned. For all the system and environment generated errors there should be a custom message or a custom error page. Never allow default error pages and messages to display in the front-end.