No direct reason is supplied for Paul's reluctance to let his mother know about his "luck." Paul says that if his mother knew "She'd stop me," and this perhaps reflects the way that Paul is gaining this knowledge. What is interesting is that a couple of lines before Paul says to his uncle that he does not want his mother to know, he is described in the following way:
The boy watched him with big blue eyes, that had an uncanny cold fire in them, and he said never a word.
The reference to the "uncanny cold fire" is something that is repeated throughout the story, and is reference to the supernatural way that Paul is gaining the knowledge about which horse will win. It hints at a kind of possession or dangerous supernatural force. Perhaps, therefore, Paul fears that if his mother knew he was lucky, she would find out how precisely he was gaining the information about the winning horses, and she would be so appalled at how Paul was risking himself, she would insist he stopped. As this is the only way Paul has to gain his mother's affection, he cannot possibly let her know about his "luck." The rocking-horse is Paul's only hope of gaining his mother's affection and he will not let it be taken away from him.