The journey Paul goes on is to win his mother's love. We learn in the opening of the story that Hester, his mother,
could not feel love, no, not for anybody.
She can't even love her own children, because something is missing inside her. Nobody else realizes this, but the children do.
When Paul, her oldest, finds out that his mother feels she has no luck, he decides to provide it for her. He knows she wants more money, just as everyone in the household does. The very walls of the house seem to be saying,
There must be more money! There must be more money! The children could hear it all the time though nobody said it aloud.
Paul also feels a bit angry when he tells his mother he is lucky and she doesn't believe it. He wants to prove himself to her.
Paul learns that if he rides his rocking horse hard enough, he can discern the winner of the upcoming horse race. He speaks about this to his uncle Oscar, who learns that Bassett, the gardener, has been placing winning bets for Paul.
Paul starts winning bigger amounts of money and arranges to have all of it given to his mother without telling her who it is from. However, no matter how much he wins, it is never enough to fill the empty space in her heart. Paul rides harder and harder and finally dies from riding too hard, but not before he wins her a huge amount of money in the Derby.
Paul, sadly, is too young and innocent to know that no amount of money will ever win him the thing he most desires. He is unsuccessful in his quest, for which he sacrifices everything, and we feel sad for him in his longing to have what every child wants but which money can't buy: a mother's love.