If only Sky Sports News only works on your smartphone NowTV App and Sky Sports Premier League, Action, Arena or Racing channels don't work then have you used a day or week sports pass in the last 30 days or have an active entertainment, cinema, kids or hayu pass ?
If so, then it sounds to me that either your sports mobile pass purchase didn't go through or you have another account when you purchased the mobile pass.
When you go to My Account > Passes & Vouchers on here, together with Bills & Payments does it indicate your mobile sports pass ?
Should the mobile pass be showing, maybe delete the NowTV App off your phone, re install the App again and re sign back into the App.
With your mobile sports pass, you should have 5 number available Sky Sports live channels to watch, which are Premier League, Action, Arena, Racing & News.
I don't know why you can watch Premier League, but the other above listed channels don't play on your smartphone screen (casting not permitted).
See if live chat can assist you further.