I believe the phrase that best describes Hughie Erskine in the first paragraph of "The Model Millionaire" is

he had every accomplishment except that of making money.

This best describes Hughie because it sums up both his predicament and his personality. We learn that he is handsome and well-liked by both men and women. He is pleasant, as he had never said an "ill-natured" thing, and not terribly intelligent. We see he has put some small effort into finding a career but does not have the personality or temperament for the daily grind in finance or business. He receives a small allowance from an aunt, enough to keep him in at least barely decent circumstances as a single person, but not enough to get married, which is his problem as the story opens

The phrase is somewhat tongue-in-cheek or humorous because Hughie is hardly "accomplished" in the sense of having a set of skills. But he does truly have what to Wilde are the most important traits, as the story will go on to show, of empathy, generosity, and compassion. He genuinely does feel for and care about the people that cross his path. When he realizes that his friend, Alan Trevor, has been painting a man who seems to be a beggar, the good-hearted Hughie gives him money, not knowing the "beggar" is really a wealthy man.