Nah this is wrong, Mancini's 2 contracts, the one with the club to be manager and the one with Al Jazira were both signed on the same day.
IMO It's pretty clear they were using the Al Jazira job as a way of reducing Mancini's salary on the City books. For starters, City were only paying him £1.4m per year to be manager which is less than half what Mark Hughes was reportedly paid (£3m) by the Shinawatra regime.
And then there's the email (We have some payments that require to be made by Al Jazira...We will need to send monies to ADUG and ADUG will then pass on to Al Jazira with payment instruction.) which intimates money is going from City to Al Jazira. What reason do Manchester City have to send money to Al Jazira?
The real question is not whether it happened, it's whether it was actually against the rules to do so.