The 61-year-old, who completed a second spell in charge of Republic of Ireland prior to taking up the APOEL job, has seen his side struggle in recent matches.
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Real Madrid entered the match having already qualified for the round of 16 as Group H runners-up, but Dortmund needed a favour from section winners Tottenham Hotspur in a bid to secure third position ahead of APOEL
The Cypriot champions had their moments in Nicosia but three goals from Kane in a 28-minute period either side of half time ensured that Spurs – as well as Real Madrid – opened up a six-point gap at the top of the group
Ronaldo netted once in either half at the Bernabeu, while Sergio Ramos was also on the scoresheet during a comfortable night for the reigning European champions.