Q: iCloud backup doesn't include recent emails

It is my understanding (maybe I'm wrong and someone can correct me, if so) that an iCloud or iTunes backup saves your email settings, but not the email messages themselves.  You should be able to download the actual mails from your server. Try:

Settings > Mail, Contacts, Calendars > Fetch New Data > (and the Fetch setting for the specific mail account)

You are correct, javaliga

Here's the official Apple explanation of what's included in iDevice backups:

About backups in iCloud and iTunes - Apple Support

Yes, I read that article.  It includes lists of what is NOT included in backups, but the lists don't include emails.  Why not?  It just doesn't say anything explicitly about emails at all.  I inferred that emails weren't saved in my response.