Thunderbird keeps track of messages it has retrieved from the server using the popstate.dat file. When you delete that message, then undelete it, it's still the exact same message with the exact same message ID as far as Thunderbird is concerned, and it's already been tracked as retrieved as stored in that popstate.dat file, so Thunderbird won't retrieve it again as if it was a new message. This is by design, unless I've misunderstood what you mean by undelete.

You understood it correctly. This design does not work if only the message header is retrieved. Should I delete the record from the popstate.dat file?

You shouldn't edit that file, but if you feel much inclined, go ahead. I suggest using the Archive option instead of Delete. That way, you can declutter the Inbox while still keeping those messages at hand for later review, without affecting the server copy. It's less of a hassle and safer than deleting/undeleting. Alternatively, you can uncheck the option "Until I delete them" in Account Settings > Server Settings > Leave messages on server so that the server copy is not deleted when you delete your local copy. This way, you should be able to download the rest of the message because the server copy is still intact and accessible in the server's Inbox, even if the message (header) has been moved to the local Deleted folder.

I will try what you suggested. But based on your explanation of the popstate.dat file I think that Thunderbird should delete the message information from the popstate.dat file whenever it performs the undelete function.

It wouldn't work because Thunderbird does not send an "undelete" command to the server via POP. The undelete action is purely a local operation. It is not replicated on the server, unlike the delete action which is replicated on the server. This is a limitation of POP, not Thunderbird. For a fully synchronised experience, you should use IMAP instead.

I am sticking with POP for economic reasons because the messages pile up on the server. I have both an iPhone and an iPad which both use IMap. I keep POP on my desk computer because I have control of the messages. If the issue occurs again I will either edit the popstate.dat file, delete it, or login to the server and send the message to myself.