Charlotte wants Lucy to confess because she has already endangered the integrity of the secret and does not want to be held accountable for it. Things between the cousins have been very difficult, and a good part of it has to do with Charlotte's interference in Florence. Although Lucy is not self-aware enough to realize the origin of her anger, Forster hints throughout that Lucy feels a great deal of resentment towards Charlotte. Charlotte represents a woman who never realized her potential. From here until the end of the novel, Forster begins to set up parallels between Charlotte and Lucy.
Charlotte represents failed potential in different ways.
Because this is a social comedy and, by the formula of the genre, the novel will end in marriage, spinsters are usually seen as somewhat pathetic figures. In A Room with a View , a life of companionship is seen as better than a life alone. Forster, newly involved with his first great love at the time this novel was written, depicts a world where life partnership can and should be part of growth. She is unimaginative and passive aggressive, and her years alone have only exacerbated these qualities. Lucy's slip is one of the nicest moments of the novel.