The 25-year-old Cano has had a trying season, batting .260 with 13 home runs and 61 RBIs. His plate discipline has seemed to regress after brief flashes of improvement, and Cano's alertness and aggressiveness defensively have not been optimal at all times.
He appeared in Monday's victory as an eighth-inning pinch-runner and said he had no issue with being benched for two games.
"He's right," Cano said of Girardi. "I made a mistake and I paid for it."
Cano was punished for the play on Sunday, which tipped Jason Giambi's glove and shot into right-center field unpursued as Floyd hustled it into a double, but it was not the only play that Cano might have benefited from being more aggressive on.
He did not chase a game-winning hit on Aug. 10 at Anaheim because he thought first baseman Wilson Betemit would get it, and Cano was a bit too smooth on a feed to shortstop Derek Jeter in a loss to the Blue Jays on Aug. 30.
"We believe that there is a lot more in there," Girardi said.
Cano acknowledged that better defense was a topic of his conversation with Girardi.
"He told me that we don't only want to win games with offense," Cano said. "Even if I don't do it with offense, I've got to play defense. He knows I'm not going to get the big hit or make the play all the time, but he wants me to show my teammates that I want to play."
Cano has just three hits in his past 20 at-bats and has not had much success to speak of this year, the first season of a four-year, $30 million contract with New York. Cano has said that he plays hard "99 percent of the time" and was "embarrassed" by being pulled in the fifth inning on Sunday.
"I would say, from now on, I'm going to forget these last two days and just play hard," Cano said. "I want to say that it's good it happened now, early in my career. I'll keep moving forward."
Girardi said that multiple Yankees players have put an arm around Cano, understanding his frustration, including Jeter. The manager said he believes Cano is "ready to be great again."
"There's a lot of people pulling for Robbie," Girardi said. "We all want Robbie to be the player he is capable of being."