Why is the King worried about the missing $415 and how is this problem resolved in The Adventures of Huckleberry Finn?

Since the will of Mr. Wilks states that the inheritance is $6000, the king becomes anxious after they are sent to the cellar to retrieve it and the amount is $415 short. Worried that they will be under suspicion when the amount stated in the will is not there, they place money of their own in the bag in order to seem above reproach. 

When the dauphin ties up the boat one day after he and the duke swindle over four hundred dollars from the townspeople who have attended their performances of the Nonesuch, a loquacious young minister named Blodgett informs the dauphin that he is waiting for two of his relatives, Harvey and William Wilks, who are due to arrive from England. As he continues, the minister reveals that the deceased Mr. Wilks left "three or four thousand in cash hid up som'ers." Hearing this, the dauphin becomes very interested in Wilks, and he relates what he has heard to the king.

In Chapter XXV, with the information gleaned from the Reverend Blodgett, the dauphin and the king, posing as the Wilks brothers, arrive at the Wilks' place. They are greeted by some of the friends and associates of the family, as well as by Mary Jane Wilks, who soon shows them the letter left by their father. In this letter, the father has bequeathed the house and three thousand dollars to the girls and three thousand to Harvey and Wiliam. Mary Jane tells them that the money is hidden in the cellar.

These two frauds said they'd go and fetch it up, and have everything square and above-board; and told me to come with a candle. (Ch.25)

After they locate the money in a bag, the two swindlers delight in running the gold pieces through their fingers. Then the king insists that they count it out. After doing this, they realize that the supposed amount is short four hundred and fifteen dollars. So, in order to avoid any suspicion, the duke suggests that they use their takings from the Royal Nonesuch and put the $415 in the bag to make the $6000 that was stated in the will. 

The two frauds climb the stairs and make a show of giving the $6000 to Mary Jane Wilks. Tearfully, she and her sisters Joanner and Susan hug each other as they cry. From the group of others who have come as friends of the Wilks family, a Dr. Robinson approaches the girls and tells them not to trust the two men who pose as Mr. Wilks's brothers; they are impostors who throw out names and facts that they have simply picked up somewhere. But Mary Jane will not listen; instead, she defiantly takes the bag of money and places it into the hands of the king, saying, 

"Take this six thousand dollars, and invest it for me and my sisters any way you want to, and don't give us no receipt for it." (Ch.25)

Angered by what he believes is foolishness, the doctor declares self-righteously, 

"All right, I wash my hands of the matter. But I warn you all that a time's coming when you're going to feel sick whenever you think of this day." (Ch.25)

Disgusted by their swindle, Huck later retaliates against the duke and the king by sneaking into their room and stealing the bag of money. Then he hides this bag in the coffin of Mr. Wilks, but is dismayed when he returns for it and discovers the mortician sealing the coffin shut.

After the king, duke, and he depart, Huck vows to write to Mary Jane and reveal the truth about the two swindlers, as well as where the money has been hidden.

The King is worried that if the gold is short, people will think he and the Duke stole it, so to avoid suspicion, they use their own money to make up the difference, planning to swindle even more money from the trusting family later.

Masquerading as the British brothers of the recently deceased Peter Wilkes, the King and the Duke learn that Wilkes has left his brothers $6000 in gold, which is hidden in the cellar.  When they fetch the gold and find it short, they replace the money with their own, then, to impress the family by their generosity and gain their unquestioning respect, they give the total amount to the dead man's daughters.  As anticipated, the daughters are completely grateful, and later, when the family doctor comes and accuses the two scoundrels of being frauds, Mary Jane, the oldest, demonstrates her total faith in the King and the Duke.  She returns the money to them, and instructs them, "Take this six thousand dollars, and invest for me and my sisters any way you want to, and don't give us no receipt for it" (Chapter XXV).