For example, 'apps gone free' would say 'your new apps gone free are ready' at about 6 PM every evening and my BBC news app would put messages on the screen and make the BBC news sound when a new piece of news came out, that no longer happens on my i Phone 6, although I believe I still have the same settings as on the i Phone 5s. I believe that is where you will find the "fetch new mail" option, although it may say something else.Sorry I can't be of any more help to you, but I've never encountered that particular problem.
So which accounts should you connect to your phone?
I uninstalled one of my Google accounts and reinstalled it and now it won't even give me a sync calendar option.
We hear from a lot of folks who have trouble syncing their i Phone Contacts with their Google Contacts.
I am using Outlook on an exchange server and when I had my i Phone 5s, mail would happily update all the time i.e.
I could hear the harp sound I assigned to mail 'doing its stuff' all the time with the new mail being added and me hearing the change in the number of new e-mails being displayed on the mail icon.