After a scoreless first leg in Belfast, Cliftonville held the hosts until McGovern's second-half dismissal.
A straight red card for a foul was followed by Rene Mihelic netting from the resulting free-kick on 55 minutes.
Ibrahima Sidibe added a second for Hungarian champions Debrecen after 79 minutes.
Cliftonville impressed in the first leg but they were missing prolific striker Joe Gormley because of a knee injury.
The Hungarians made a strong start at Nagyerdei Stadion and Dalibor Volas hit the woodwork in the ninth minute.
The Irish League side rarely threatened although Stephen Garrett and Johnny Flynn forced Nehad Novakovic into action.
Conor Devlin was the busier keeper and he twice denied Adam Bodi before keeping out a Jozsef Varga effort.
Bodi saw another strike saved by Devlin before McGovern was dismissed on 54 minutes after a challenge on Volas.
Mihelic stepped up to score from the free-kick and Volas wasted a good opportunity to double the lead three minutes later.
Jude Winchester had a shot blocked as the Reds went in search of an equaliser but hopes of a comeback faded after Sidibe's goal 11 minutes from time.
Debrecen now face Belarusian side FC Bate Borisov in the third qualifying round.
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