If Egress is stuck sending emails then the below can be used as a temporary fix.
To apply this fix we will have to use PowerShell.
Firstly, connect to exchange with the following:
Connect-ExchangeOnline
If you get an error connecting then the exchange module will need to be installed:
Import-Module ExchangeOnlineManagement
Once connected run the below to apply the fix.
Set-CASMailbox -Identity Insert user email -OwaMailboxPolicy EgressOnSendPolicy