Myrtle insists on saying Daisy's name, again and again, even after Tom has apparently told her to stop.
"Making a short deft movement, Tom Buchanan broke her nose with his open hand."
He breaks his mistress's nose for speaking his wife's name. This shows that Tom views Myrtle much differently from the way he views his wife. He's elitist, and so he treats his wife, who is of much higher worth and value, much better than he treats his mistress, who is of a lower social class and is relatively poor. Daisy is indispensable to him while Myrtle is very much replaceable. There will always be women lining up to be with Tom because he's so rich and young and handsome. Back in Chapter One, when Daisy insisted on repeating a word—"hulking"—that Tom doesn't like, he merely "object[s] [...] crossly" at her use of the word. He doesn't slap her or even get angry at her. This episode shows, quickly, how ingrained class differences are in Tom's mentality; he doesn't seem to think twice—or at all—before striking Myrtle because she is so far beneath him.
At a party in their apartment, Myrtle keeps saying the name "Daisy" over and over. Tom hits Myrtle across the face, breaking her nose. This is the second time that Tom's penchant for hurting women is very real. When Nick first meet Daisy, she is complaining that Tom is a "brute" and he hurt her little finger. However, the incident was only alluded to by Daisy. In Chapter Two, Tom's explosive anger is actually revealed as Nick and the others in the apartment see him hit Myrtle in order to stop her taunting.
Tom breaks Myrtle's nose and then smooths things over by buying her the puppy she had wanted earlier. This is an example of class and power struggle. Tom has more money and status and is therefore able to get away with things like that. Myrtle is powerless to speak out against this exploitation because of her lower status and lack of wealth.
I feel as though Tom is ashamed of being unfaithful even though he comes off like it's not big deal. Why else would he act the way he did. I think he hit her to stop her from saying Daisy's name because he didn't wanted to be reminded of his affair. Maybe..