It sounds like there might be a disparity between how the iphone are checking your email. The two most common protocols are POP (Post Office Protocol) and IMAP (Internet Message Access Protocol).
POP acts kind of like your physical mailbox at home: your mail person delivers your message, you run out to the box and retrieve the message. Your mailbox is now empty, but you (or your device) has retained the message.
IMAP works a little differently. Your mail person drops off the message, but you don't need to go pick it up - you can read it from afar. A benefit of this is that you are able to get persistence across all of your devices, as each one acts like a window into your mailbox, where your messages aren't deleted, and are made available for any device you have.
It's very possible that one of these accounts is set up incorrectly, or needs a few more settings tweaks to get your mail behaving normally. If you can give our support team a call, we'd be happy to take a look and determine which device needs some extra attention, and get it set up so that there's some consistency between your iPhone and desktop.
If you'd rather take a stab at it yourself, we have a guide on setting up Apple Mail on your desktop, which should be fairly similar to the iPhone configuration: https://help.sonic.com/hc/en-us/article ... Apple-Mail