I think there is a need for some perspective here. There is no such thing as a perfect manager who will get things right every time. Mancini makes mistakes sometimes, but so do all the others.
Those who call for Mancini's exit frequently suggest Mourinho and Klopp as ideal replacements. They are both very accomplished managers with much to offer. However, Mourinho's Real Madrid team has lost five league matches so far this season and sit 16 points adrift of the leaders. Klopp's Borussia Dortmund have lost four league matches so far including a 1-4 thumping at home to Hamburg SV yesterday. They are now 15 points adrift of leaders Bayern Munchen. I guess their fans are about as happy about that as we are about the Southampton result. So we need to keep reality in mind and forget about notions of hiring some mythical miracle worker to replace Mancini.
Mancini isn't perfect. His man-management skills aren't the best. He appears quick to diss his players in public and sometimes slow to praise when appropriate. In certain cases, individuals seem to be harshly frozen out (eg. Lescott). And sometimes Mancini needs to learn to bite his lip. By publicly lambasting Hart yesterday, he risks alienating a player who has produced countless clean sheets for the club and on occasion rescued his boss from abject humiliation (eg. Borussia Dortmund at the Etihad). So Mancini himself - as well as the players - has areas of self-improvement to work on too. But falling short of 100% perfection doesn't mean he should be discarded.
Mancini has managed City to three trophies since joining the club. He must be doing something right! And this season hasn't ended yet; City remain in contention for the FA Cup, and even a Premiership runners-up spot (if achieved) is way beyond what most of us could have dreamed of just five years ago. Life as a blue ain't so bad these days!
Perhaps we need to work with what we've got rather than look to knee-jerk change. Maybe Txiki and Ferran can encourage RM to work more empathically with his players and praise them more readily, and to avoid emotional criticism of them in the media. This is his failing, but it need not be beyond repair. Mancini must learn to overcome his weaknesses, just as the players must work on theirs. But whilst recognising the faults, we mustn't obsess about those in isolation. RM has brought many positives to the club - positives which have delivered three trophies and counting. It would be foolish to throw all that away in haste.
If anybody out there comes across the *perfect* manager, please do let us know. But in the meantime, remember that all the 'dream solution' names come with issues of their own. Mourinho and Klopp are awesome managers. But they lose games too.