K9 mail app or something similiar?

I had this exact same question (I recently migrated to iPhone from Android) and found no adequate responses for this question, anywhere. My use case is probably an unusual one because I use non-standard ports and protocol combinations for some of my mail accounts.

Blue Mail (https://itunes.apple.com/sg/app/blue-mail-email-mailbox/id1063729305?mt=8) works for me and it may work for you too.

Whatever alternate mail apps are available can be found by searching the App Store. iOS's own email app can handle just about any IMAP or POP account. I have 3 gmail accounts there (all set to IMAP online in gmail account settings), three other provider's IMAP accounts and one old POP account all in the iOS mail app - works fine with all of them. I do use an app called Mail+ for my employer's Office 365 business mail account just so I can keep that one separate and isolated from all my other emails (I like Mail+ better than the MS Outlook app)