Mark Hughes' men only scraped into the last eight on penalties following a 2-2 aggregate draw with Danish side Aalborg.
They have been disappointing on their travels all season and now face a tough trip to Germany to meet Hamburg in the first leg of their tie.
Martin Jol's Bundesliga high-flyers, who had to come from behind to beat Galatasaray 3-2 on Thursday night, will then come to Eastlands for the second leg.
Elsewhere, Werder Bremen will take on Udinese and Paris St Germain will lock horns with Dynamo Kiev.
There is also a battle between two UEFA Champions League dropouts as Tottenham's conquerors Shakhtar Donetsk play Marseille.
The draw for the semi-finals has also been made, with Man City tackling either Werder Bremen or Udinese if they overcome Hamburg.
Uefa Cup quarter-final drawHamburg v Manchester City
Paris St Germain v Dynamo Kiev
Shakhtar Donetsk v Marseille
Werder Bremen v Udinese
Ties to be played on 9th April and 16th April.
Uefa Cup semi-final drawWerder Bremen or Udinese v Hamburg or Manchester City
Paris St Germain or Dynamo Kiev v Shakhtar Donetsk v Marseille
Ties to be played on 30th April and 7th May.