By default HelpSpot does include the entire body of the note in the email notifications. It sounds like maybe somebody modified the email template to not include all of it.
That can be changed back in admin-email templates-staff notifications. Edit the HTML and Text versions of that template and make sure they include the ##message## tag. That will be the full text.
You can also include the entire public note thread if you like on each notification. The placeholder that does that is available in the drop down below the email template editing box.