Please use the below command to check the authentication methods you have configured for EWS.
Get-WebServicesVirtualDirectory | fl *auth*
According to the error message, we can try to recreate EWS and OWA virtual directories to test.
In addition, I have found a similar thread for your reference: https://social.technet.microsoft.com/Forums/exchange/en-US/1c5758d8-bf2c-4b24-8959-1bdcb9734b75/owa-2010-unable-to-delete-emails?forum=exchange2010