In wp-admin/includes/update.php
file
if ( current_user_can('update_core') )
$msg = sprintf( __('An automated WordPress update has failed to complete - <a href="https://wordpress.stackexchange.com/questions/231010/%s">please attempt the update again now</a>.'), 'update-core.php' );
else
$msg = __('An automated WordPress update has failed to complete! Please notify the site administrator.');
We can see that messages are different based on the current user role and this is maintenance_nag
.
Basically we have two update nags and can be found in admin-filters.php
add_action( 'admin_notices', 'update_nag', 3 );
add_action( 'admin_notices', 'maintenance_nag', 10 );
So to remove second message we can use(also check for current user role if you want this only for non-admins)
remove_action( 'admin_notices', 'maintenance_nag', 10 );
For multi-site use
remove_action( 'network_admin_notices', 'maintenance_nag', 10 );