Answer: pleading
To beesech is to ask for something urgently and repeatedly. In this excerpt, Huck is mad at his aunt. He can tell that she wants to apologize to him, but cannot not bring herself to do so. Huck believes that his aunt will eventually run to him, in his deathbed, and implore him to forgive her. However, he takes pleasure on the idea of her being so desperate and Huck never, ever forgiving her.