You can test if Outlook 2016 is working properly by going into Windows Explorer, find any small file, right-click on it then choose Send To -> Mail Recipient. It should open Outlook 2016 as a draft email and the file attached. If it opens a different version of Outlook, Microsoft Mail or other email client then it's set up wrong.
It seems that Outlook 2016 has a problem with certain Sage products, Quickbooks and other programs that use MAPI and MAILTO functions i.e. any program that asks Outlook to perform emailing. Earlier versions of Outlook work fine as do other email client programs.
TPN Invoicing uses MAPI to send emails. If you cannot email out of TPN Invoicing using Outlook 2016 then got into Control Panel/Default Programs and choose the button that says 'Reset to Microsoft recommended defaults'.
Run Outlook as an Administrator and go to the Trust Centre. Find the Programmatic Access Security and set the 'Never warn me about suspicious activity'. Exit Outlook.
Test a simple function such as generating a CSV invoice. This should now work.
You should seek any further technical assistance from your original supplier of Office 365/Outlook 2016.
Norman