Are there notifications in Core for managers when time is not submitted or the staff is not completing their time cards?
You are looking for a functionality that notifies your employees and managers about missing or incomplete time cards. Currently, Core does not have the ability to notify when time is not submitted or filled in. However, you can schedule a time and expense report filtered by workflow status 'Unsubmitted' and have Core deliver it to you (manager or supervisor) on a daily or weekly basis. This assumes that the time is entered in the system and just needs to be submitted to you. Check the Core Help for details.
Your employees can turn on a reminder in User Preferences (My time needs to be submitted) to help them remind to submit time. Check the Core Help for details.