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Maybe you accidently enabled "Restrict Background Data?
Try this
Go to Settings > Data Usage > Press the "Three Dots" on the upper right corner >Untick "Restrict Background Data" .Then restart phone.
Also,don't forget to add your apps to Stamina Mode's White List(that enables you to receive notification of specific apps you added into white list even your phone is asleep)
Hope that helps
Hi nz440v8
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Have you tried performing a settings download, it could possibly be the wrong settings Internet wise.
Menu > Settings > More > Internet Settings > Accept disclaimer > Choose network carrier.
Once done try using the apps.
let us know how you get on.
As suggested by Jake-SonyXperia
If this doesn't help it may be your operators proxy-server causing issues for certain connections. Try removing proxy/port from your settings under Settings -> More -> Mobile networks -> Access Point Names -> [your settings] and see if that helps.