Why won't I receive certain emails on my iPhone?

I recently got the new iPhone 5c. I connected my Hotmail account into the "mail" app. I can only receive certain emails, and what I mean by that is that I hear the "ding" sound and I look on the icon of the app and there's a little red "one" on the top right of the icon, but it happens very rarely. I can't receive most of the emails, like it won't "ding" automatically and no red "one" appears next to the mail app. To view & get my emails, I need to go on the app, and it will refresh on its own, and I get all my emails then, but I would prefer if it would do so automatically, like without me going on the app. Everything was perfect on my iPod, I got all my emails automatically, like I would be on Facebook for example, I hear a "ding" saying I got an email. Now I disconnected all my accounts on my iPod so it won't interfere with my phone, so having 2 same accounts open is not the problem, because my Hotmail is only open on my phone now. Should I call apple about this? Because in my settings I set it to get email automatically, or manually as some might call it. So having the wrong setting is not the answer either. Thank you for your help.

And please do not say I need to restart my phone, delete the account and re-connect it, because trust me, I ALREADY HAVE! LOTS OF TIMES!

Unfortunately, there isn't much you can do at this point. This might sound cliche, but all you can do now is factory reset it with itunes.