Champions City, now seven points behind leaders Manchester United, lamented another wasted trip to Sunderland after also having lost there in the league on January 1.
"Maybe the referee ate too much for Christmas," said Mancini before adding that Friend missed a foul on Pablo Zabaleta in the build-up to the 53rd-minute winning goal by City discard Adam Johnson.
"You need to ask the referee why it was given, not me. I saw it was a big foul - it's impossible two men [Friend and his linesman] couldn't see it," added the Italian.
"Maybe next year we won't come here. We won't spend the money," he told City's website in a tonque-in-cheek remark.
The visitors enjoyed the better of the first half with Vincent Kompany hitting the bar and Sunderland keeper Simon Mignolet making good saves from Yaya Toure and David Silva.
After Johnson's second-half effort beat keeper Joe Hart, City struggled to create clear chances and Sunderland held on for a victory that lifted them seven points clear of the drop zone.
"We need to score... our strikers need to improve," Mancini said. "We need to change something in our team because we can't continue like this - we score too few goals.
"Sometimes we take too many touches and we think, 'never mind, we can score next time,' but we can't be like that - in football it doesn't work like that. We have to be stronger in the penalty area."
Second-placed City next play at Norwich City on Saturday and host Stoke City on January 1.
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