Hi @Andy Hodkinson ,
Welcome to our forum,
I have researched a lot about your needs but haven't found a direct method to manage the mailbox's task via exchange powershell. However, I found a related script that mentioned getting Outlook Tasks in PowerShell, please kindly check if it could be helpful to you.(Please Note: Since the web site is not hosted by Microsoft, the link may change without notice. Microsoft does not guarantee the accuracy of this information.)
If that's invalid, please kindly understand here we mainly focus on general issue about Outlook desktop client, considering your issue may be more related to powershell script, in order to solve it better, it's suggested to add other powershell or development tag in your post, so that there may be some relevant experts to provide you with more targeted suggestions on powershell or development.
Hope your issue would be resolved soon.
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