I will try to put all the information you asked for in this post.
To get the Igadzra string, you must have gone to Mira(in the Mirava system), and taken a Miranu message to Earth. After that, you must go to the Blaga system, and join the Miranu Trade Conglomerate(MTC) via a bar mission. Upon the completion of those, you must go to Kitrak with a Noteworthy combat rating.
If you have not been told that you have completed one of Override's Central Objectives, then you are not done with the Igadzra. One thing that may be holding you back, is a combat rating lower than 'Ultimate'. If you have that rating, then just keep checking Igadzra's Bar.
You will know you have completed the Igadzra string when you receive a message saying:
"Congratulations, <PlayerName>. <Spoilers> You have completed one of Override's central objectives, the destabilising of the Strand War. Now you can but wonder what effect this tilting of the balance will have."
That should be all you asked for.