Left wing/right wing has nothing to do with it. Those terms were never anything more than shorthands. They were never desgned to pigeonhole or entrench.
When it comes to the Fourth Estate's treatment of City, the truth is far more simpler and more prosaic. Far uglier too.
There's only one left-leaning national newspaper in the UK and that's the Guardian. Admittedly, they do hate City's owners. This is because of perceived connections to human rights abuses. They're hypocritical in their own way but, for the sake of simplicity, let's park that for now and deal with the larger phalanx of hate.
There are several famously right-leaning publications such as the Times, The Telegraph, The Sun, The Star and The Daily Mail. Others like the Evening Standard, the Independent and the Daily Mirror are less far over on the spectrum but still conventional and usually right of centre (although the Mirror flip flops).
All of these publications like to pronounce that ultimately they hate City's owners because of perceived human rights abuses. They don't. Let me explain why.
There's no à la carte menu of human rights.You can't pretend to be concerned about human rights abuses and champion the Tories current attempt to simultaneously drive a coach and four through the UK Courts' decisions and the European Court of Human Rights declarations.
Whilst the individuals working for them might have their own agendas when it comes to City, overall, it is my opinion that primarily these publications give them free reign to spout bile because they hate our owners based in the colour of their skin. (And their religion too, probably).
Nothing we ever do or achieve will win over their racist hearts.
TL/DR, they're all horrible organs.