This article explains a few solutions on how to retrieve a deleted Facebook Messenger message. These include viewing your archived messages, downloading your Facebook data in the hope that your message is still on the server, and asking your contact for a copy of the conversation.
How to Recover Deleted Messages on Messenger (Update) | Retrieve Deleted Messages
An archived message in Messenger is hidden in your inbox, but it’s not permanently deleted. You may have selected Archive instead of Delete. It’s a mistake you easily make, and you’ll be glad you did, because finding archived Facebook messages is easy.
One way to find archived conversations is through the Messenger app. Here's how it works on the Android app (the instructions should be similar on an iPhone):
Open Messenger and tap the three-line menu at the top.