Shamim's Plugins

Those are the settings. Do you have any caching setup? If yes, purge all caching.
If your website is open in some browsers, this call will continue to make until they close the browser or navigate to another page.

You can send email to admin using following code. Change content as you like.

add_action( 'fep_action_message_after_sent', function( $message_id, $message, $new_message ){
	wp_mail('admin@example.com', 'New Message', 'New message waiting for approval' );
}, 10, 3);

Is the message sent to Admin or any user?

Plugin customization is not included in paid version. after that also i try to provide code if it needs small code snippet.
But to achieve your required functionality it not only needs template editing but backed code as well.

If you know php you can follow following steps
1. Edit view-message-content.php template file to show select field of users with search capability.
2. Add an admin ajax hook or backend function to handle selected user
3. Set that user as participant of that message using insert_participants method

Please contact using https://www.shamimsplugins.com/hire/

Did you already added CC field there (I see in your previous message)? If yes, how you added that?

If no, do you want to give user an option to add cc there? Or you want to add automatically? If automatically, what is there user id which you want to add as a CC for all messages? So that i can write some code and give that to you which will automatically add that user to all messages for you.

You can do any of the following.

1. You can set any email address you want in Front End PM PRO > Settings > Email > From Email. So when user receive an email, they will receive from this email. You can set an auto reply in that email to notify users that they will need to send message in your website

2. You can set Email Piping (https://www.shamimsplugins.com/docs/front-end-pm-pro/getting-started-2/email-piping/) Or POP3 (https://www.shamimsplugins.com/docs/front-end-pm-pro/getting-started-2/pop3-feature/) so that if user reply to any email, that will automatically send message in your website.

3. Disable email notification.

It should be one time change.
Do you have any other plugin which may have cron change functionality?