Phil Hughes allowed three runs and six hits in six and two-thirds innings.  KANSAS CITY, Mo. — After a particularly dispiriting loss Saturday night, Alex Rodriguez set out the core expectations that he and his teammates in the middle of the batting order hold sacred: damage, production, wins.  In a game the Yankees felt they had to win, they followed that formula rigorously and came away with a 10-4 victory over the Kansas City Royals on Sunday to salvage a split in a series that was a disaster even before the first game was played.  Closer Mariano Rivera was lost for the season after he tore a knee ligament while chasing a fly ball in batting practice Thursday. The Yankees lost that game, and Saturday’s, too, as their offense continued to sputter.  But in the finale Sunday, the offense returned with authority. Robinson Cano, the previously struggling No. 3 hitter, did the most damage with a grand slam in the Yankees’ six-run third. The productivity came from Nick Swisher, who homered in his first game back from a hamstring injury, and from Rodriguez, who blasted a three-run homer for insurance in the eighth inning.  Phil Hughes took care of the win, delivering his best outing of the season to improve to 2-4. The victory ensured that the Yankees, who said goodbye to Rivera before Saturday’s game, avoided a third loss to a Royals team that before Thursday had not won a home game all season.  “It’s huge,” Rodriguez said. “We couldn’t afford another loss today. We definitely came out and played with a lot better energy.”  The first four batters in the lineup — Derek Jeter, Curtis Granderson, Rodriguez and Cano — each had two hits and combined to drive in eight runs. The Yankees have struggled with runners in scoring position recently, but went 4 for 11 in those situations Sunday.  The Royals scored first off Hughes, but the Yankees tied it in the second on a run-scoring single from Raul Ibanez. The Yankees piled on six more runs in the third as Cano and Swisher homered off starter Luke Hochevar (2-3).  Cano, whose average dropped to .255 after a 1-for-4 showing Saturday, said he refused to be frustrated. He said he had had worse struggles, pointing to the 2008 season, when he hit .271, and said he was confident he would break out in time. With one swing 14 hours later, Cano matched his home run and R.B.I. totals for the season.  “I was even worse in ’08,” he said, “And I came back and hit .270. At this point I was at like .150 and .160. In this situation, I’ve been hitting the ball good, just right at people.”  Hughes made all that offense stand up. He pitched into the seventh inning with the best fastball he has thrown all season: hard and true. It was clocked around 95 miles per hour, even into his final inning, and his command was not questionable, as it had been in his previous starts.  Manager Joe Girardi was particularly impressed with Hughes’s changeup, a pitch he has been trying to master. Hughes threw several good ones, including an audacious one on a 3-2 count to Mitch Maier, who struck out with a runner on and one out in the fourth.  In submitting his best outing of the year, Hughes probably ensured that he would retain his spot in the starting rotation for now. Andy Pettitte pitched again for Class AAA Scranton/Wilkes-Barre on Sunday in what may be his last rehabilitation start.  “Our plans are that guys are on turn,” Girardi said. “That’s the bottom line. I’ll continue to put out the rotation as we go forward, but he pitched great today.”  One of the two highlights for the Royals was when 28-year-old rookie Irving Falu tripled in his first major league at-bat. Falu spent the previous nine years in the minors, including 376 games at Class AAA Omaha, before being called up for this series. Humberto Quintero homered off Hughes in the seventh.  Other than that, it was the Yankees with the damage, the production and, thanks in large part to the middle of the lineup, the win.  “In order for us to do what we want to do,” Rodriguez said, “Robbie and I have to swing the bat much more like we did today.”  Mark Teixeira’s case of bronchitis has not gotten better, and his coughing fits have him worried about straining a muscle as his whole upper body contorts. Teixeira said he was hoping a new medication would help. ... Curtis Granderson collected his 1,000th hit, a single in the third inning. ... Mariano Rivera is expected to visit specialists in New York on Monday and Tuesday. ... Andy Pettitte pitched five shaky innings at Class AAA on Sunday, taking the loss in a 7-5 defeat to Pawtucket. Pettitte gave up five runs, three earned, and eight hits. He walked two and struck out five.
Phil Hughes pitched perhaps his best game of the season, and the Yankees erupted with three home runs, including a grand slam by Robinson Cano.