You can have your conversation and everything in it removed. This page is also the deletion route required by Facebook for apps that use Messenger.
Write to the address on the website you were using from the email address you used, or naming the phone number you texted from, and ask for your conversation to be deleted.
Deleting a conversation destroys its messages and every image in it, including the files on disk. It cannot be undone and there is no copy kept for later.
Two things deliberately survive it:
Write to the address on the website you were using and name the Facebook Page you messaged. We will delete the conversation, the messages in it and any attachments, together with the page-scoped ID, name and profile picture we received from Meta for you.
A business can also disconnect a Page at any time from the application's settings. That unsubscribes this service from the Page immediately, so no further Messenger data is received; conversations already received are deleted on request as above.
This page is the data deletion instructions URL for our Meta app.
An owner can delete any conversation from Management → Conversations, which removes it and its images at once. To remove the whole workspace, write to the address on the website you were using.
Requests are actioned within 30 days, and usually within a few working days.