Mancini had a consultancy role with Al Jazira FC prior to joining City, it was discussed on here at the time he was appointed (Dec. 2009). When he joined City he signed the new contract with Al Jazira FC that you referred to.
A couple of people have said this, but there is absolutely no evidence of it anywhere. Not a single web result for Mancini and Al Jazira exists until the Der Spiegel accusation.
Now the internet wasn’t as saturated in 2009-2019 as it is today, but if a major manager reported to be working for a club it would be online somewhere.