Email big annoyance

OK I've been trying and trying and trying to fix this problem but it seems it just seems crap on android.
using a HTC ONE A9

What's the problem :- proper push email.

THIS IS WHAT I WANT!

I want an email app that can push email like my wifes Iphone 4 BUT and this is a BIG BUT I want a tick box that says keep email on server. If I delete it on my phone I want the option to have a copy still on hotmail.com. (POP3). It works on my wifes Iphone.

Why not on my HTC?

The HTC mail app is crap I have the option to use the outlook.com widget which gives me PUSH Mail, but it doesn't have keep mail on server tick box and deletes it on the server when I delete it on my phone - grrrr

I setup again using pop3/imap widget but doesn't have push!

I have tried other apps (k-9 - yuk), blue mail (yuk) outlook app bigger (yuk). yuk=didn't like!

I have the settings in outlook and gmail that says with pop3 keep mail on server and the phone still deletes it!

Sorry for the rant, but in this day and age of modern phones I didn't expect such difficulty.

Thanks for any advice in advance!

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