I cant' see how the word "cut" could mean anything but basically snubbing people who had been rude to her in hte past. What else could it mean?

Lucy has been a poor relation all her life and has had to conceal her feelings about being either patronised or snubbed by other people, in order to get a few decent invitations etc. But I'd say that a lot of people didn't invite her, or show her any attnetions, or if they were forced to do so, did it in a condescending spirit... and she is now rich enough to sneer at them all and talk about her rich husband and her house in London or whatever she has to boast about...