Twain's intent in showing the blackmailing scene between the constable and the woman appears to have two components. The first is to advance the structure of the story, and the second is to illustrate his point that the common people are completely at the mercy of those who are in power.
Structurally speaking, the blackmailing scene between the constable and the woman provides the means for Miles Hendon to use in securing the little king's release a little further on in the story. Miles has witnessed the exchange between the poor peasant woman and the law officer, and uses this information to engage in a little blackmail of his own in convincing the constable to let young Edward escape. Miles makes the suggestion to the constable, and threatens to tell the judge exactly what he has seen if the man does not cooperate with his scheme. Although the constable balks at doing what Miles wants at first, he is convinced when Miles explains to him the exact nature of the crime he has committed in his dealings with the poor woman, a crime for which the penalty "is death by the halter, without ransom, commutation, or benefit of clergy" (Chapter 24).
Twain also shows the blackmailing scene between the constable and the woman to further his theme of how completely the common people are at the mercy of those in power. The woman, who has exhibited deep sympathy and kindness in lowering her estimate of the value of her goods before the court so that the little king will not hang for allegedly stealing from her, is forced to sell what she has at that same low price to the unscrupulous constable who is quick to take advantage. The constable threatens to expose the woman for lying under oath in devaluing her goods, even though she only did it out of mercy for the boy accused of robbing her. The woman represents the peasant class, who are good at heart but will always find themselves helpless and at a disadvantage under the ridiculously harsh laws of the land and those who enforce them (Chapter 23).