I’ve converted to mp3 and synced to iPod for podcast. It has not worked correctly. And than I synced mp3 file for audiobook. Bingo! It has worked correctly.
I'm not sure I understand by "synced to iPod for Podcast" and "synced mp3 file for audiobook". It's one or the other and changing something to a podcast doesn't make it a Podcast (note the difference between podcast and Podcast). The same "audio file" can be assigned (labelled) by the producers as a Media Type of Audiobook and downloaded by you to your computer as an audiobook, or it can be assigned Media Type of Podcast and delivered automatically to your iTunes Library by the Podcast delivery system "as a Podcast". In both media types, the "genre" would be the same, such as "crime thriller" or "historical drama", but the method of delivery to your computer is different.
This is relevant because (in order to work out what the problem is*) it's extremely unusual for Podcasts to be in m4a format since Podcasts are designed to be universal and playable on any media player, while m4a is Apple's own format and not every media player can play them. Every audio Podcast I've ever downloaded is in mp3 format.
So is this a file delivered to your iTunes Library "as a Podcast", or is it something that the provider calls a Podcast but isn't in fact. Is it meant to be an audiobook?
Is it from Audible.com? I understand that Audible have two different formats, of which one doesn't play well in iTunes or on iPods. They don't make it clear which is which.
* Additional text added subsequently by author.