While you can’t delete a message from your LinkedIn Recruiter inbox, you can archive message threads which allows you to set aside the messages and retrieve them later.
To retrieve an archived message thread:
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Move your cursor over the Messages icon at the top of your Recruiter homepage.
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Click View all messages.
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Click the Inbox tab on the left-hand rail and select Archived from the dropdown. Your archived messages appear on the left-hand rail.