Carroll helped himself to his first two goals in a Liverpool shirt since joining from Newcastle for £35 million in the January transfer window. Dirk Kuyt also netted on a torrid night for City, who lost captain Carlos Tevez to a suspected hamstring injury ahead of Saturday’s FA Cup semi-final with neighbours Manchester United at Wembley. Mancini’s expensively-assembled side has now failed to win any of their last six Premier League away games and remain fourth in the table. The Italian is under pressure to qualify for next season’s Champions League. But City are only three points ahead of fifth-placed Tottenham, who they entertain on May 10, while their London rivals also have a game in hand as the season reaches a thrilling climax. It was hard to believe judging by their first half performance, but City had arrived at Anfield boasting more clean sheets than any other team in the Premier League. Yet hopes of a 15th were undone by a shambolic first half defensive display as Liverpool laid the foundations for a seventh win in 12 league matches under Kenny Dalglish with a dominant first half display. Five-nil winners over Sunderland in their previous game, City were given…