Robinho was guilty of the latter when he made space for himself on the left-hand side of the area but flashed a shot wide instead of trying to find the unmarked Stephen Ireland in what was City's only other opening of note.
Luckily for City, Omonia seemed more interested in keeping goals out than scoring themselves.
The Cypriot side had taken the lead in the first leg but the closest they came to beating Joe Hart while City were toiling was when an Efstathios Aloneftis corner narrowly eluded the onrushing Timo Wenzel.
And any hope that the visitors had of upsetting their hosts was extinguished by two goals in the space of seven minutes after half-time.
City had looked markedly improved when they came out after the break and it was no surprise when Elano gave them the lead, meeting Robinho's pass with a fine first-time strike from the edge of the area.
Robinho escaped down the left again seconds later and it took an important interception from Georgallides to prevent his cross reaching Ireland who would have had an easy tap-in.
Wright-Phillips brought another save from Georgallides soon after, when he met Javier Garrido's cross with an instinctive shot, but he did not have to wait long to get on the scoresheet.
He all but made sure of City's progress when he latched on to Ireland's flick, evaded two challenges via a series of mini deflections and buried a fierce shot into the back of the net.
From then on it was a procession for the home side, who were suddenly using their wide players to good effect and delivering a succession of tempting balls into the box.
Robinho came close to adding a third goal when he forced a fine near-post stop from Georgallides but it was actually the visitors who were next to find the net.
Alabi escaped Tal Ben-Haim's attention at a corner and beat Hart with a powerful near-post header to give Omonia's vocal travelling support something to celebrate.
That took some of the gloss of City's second-half display but Hughes can take some satisfaction from the way his players responded after their stuttering opening.
Another boost for Hughes was the return of Martin Petrov, who came off the bench after a month out with a hamstring injury and brought two more decent saves out of Georgallides before the end.
Manchester City boss Mark Hughes:
"I felt it was important for the development of the team that we progressed into the group stages and we have done that.
"Now we have a number of games in this competition to look forward to.
"I think we can reach the latter stages so let's hope for a decent draw and see what we can do."
Manchester City: Hart, Zabaleta, Richards, Ben Haim, Garrido, Wright-Phillips, Ireland, Kompany (Hamann 66), Elano, Robinho (Petrov 70), Jo (Evans 67). Subs not used: Schmeichel, Ball, Fernandes, Sturridge.
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