The PUT method creates new Timesheet entries. Clarizen allows multiple Timesheets entries for the same work item/resource/date and displays such information as multiple lines for the same work item.
If I understand correctly, your application is expected to update the existing Timesheet entry if one exists.
To update an existing entry with the API you need to use the POST method and specify the existing timesheet entry in the URL.
Refer to API documentation for more information.
Hope this helps,