I suspect then that will be some eg: iOS type system issue “like the iOS badge counts” that sometimes doesn’t always update but other times does due to whatever’s going on with the iOS System as @paf advised in this thread.
I did another test to my iPad this morning with eM Client and the email was there instantly and the badge updated ok, but the second time I tested it took a few seconds to show the email in the client and the badge didn’t update. The third time i tested the badge updated around 10 secs later. So just all seems to depend on what’s going on in the iOS System.
The way eM Client mobile app works with Push Mail is that, when you get an email on your server mailbox end, you get a notification banner, badge appear, and then when you press on those notifications, they load the mobile app and retrieve the email “which with the new app is now (very fast) compared to the older iOS app”.
So when you open the mobile app (not using the notification) “it also works the same way” and the new email should appear pretty quick. For myself the emails usually appear nearly instantaneous or at least within a few seconds using my 50mbps Wifi home connection.
So if when you open the mobile app you also see new mails within a few seconds then there is nothing wrong as per how it’s designed to work.