I just set up an email handler for one of our service desks. There doesn't seem to be the standard setting in the config that allows the emails to remain in the account after forwarding them to the service desk.
I set this up according to the Atlassian recommendation that it be done directly in JSD and not in JIRA mail handler.
In order for the emails to remain in the email account you will need to use IMAP as the protocol. If you use POP they will be deleted.
From Atalssian's documentation
Emails using POP
To link your email account with a service desk using POP, make sure that your email inbox is empty by moving the existing messages to another folder, archiving them, or deleting them. Starting with an empty inbox ensures that you do not lose emails unintentionally, as POP emails are deleted after they are processed by Jira Service Desk.
Emails using IMAP
If you use IMAP, emails are marked as read (not deleted) after they are processed by Jira Service Desk. If you want existing messages to be pulled in by Jira Service Desk, you can move them back to your inbox and mark them as unread after the connection has been established.
Hello Atlassian Community! I'm Tania, a Senior Product Marketing Manager for Jira Service Desk Cloud! I'm excited to announce some exciting improvements coming up for those who work with queues and...
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