When clicking Delete in the Modern UX for a custom object, a user receives error: API-1007 : You are not authorized to process request. Contact your system administrator for necessary security rights.
This is due to the user missing delete access to the object.
If a user does not have delete rights for the object, they will still see a delete option but the delete attempt will not be successful. When they attempt to delete the object, the user will get the API-1007 error.
Grant the user delete access to the object.
Classic users will not see a delete button.