Europe now represents Hughes's only hope of delivering a trophy this campaign, a feat that would not only make the club more attractive for potential summer signings but would also ease concerns that City's Emirati owners are not convinced the Welshman has a long-term future at their side.
The former Blackburn manager will hope, then, his talks with the Brazilian international in the Danish capital before the game will solve the mystery once and for all. If the rot continues, Hughes may find another man brought in to stop it come the end of the season.
He said: 'When Robinho plays well, he is a very positive influence on the team, there's no doubt about that.
'He is very important to us, but it is difficult for him away from home. The fans give home sides an edge, referees maybe give them more decisions and space is at a premium. Teams know if you give Robinho space, he will hurt you, so they have plans to deal with him.
'But our best performances away from home this season have arguably been in Europe, so we hope for an improvement. Ideally, we'd like to match the level we showed against Schalke in the group stage. That was a game that nobody expected us to win and yet we managed to come away with it.
'We know we have to improve away from home, it's something that we have acknowledged for a long time.'
While FC Copenhagen, managed by the former Wimbledon player Stale Solbakken, represent a stern test, greater challenges await in later rounds should City progress. Fellow Danes AaB Aalborg, who beat Deportivo La Coruna 3-0, are likely to be up next, but after that, the likes of AC Milan, Valencia and Fiorentina await.
Hughes said: 'There are a lot of good teams in the Uefa Cup and it's probably tougher than it has been in recent years. We've had a good run in the competition already, given how long ago we started. The aim was to get to the group stage and we managed to do that and top our group.
'Now we have to try to negotiate the next two games and then see how far we can get in the latter stages.'