How can you write an internal monologue of Mr. Cunningham when he is at the jail with the lynch mob? What is he thinking about when Scout is talking to him?

When you write an internal monologue, you have to put yourself inside the mind of a character.  This is similar to the advice Atticus Finch gives his children.  He sometimes tells them to walk or stand in another person's shoes.  
An internal monologue is the inner voice of a character.  The internal monologue can also be thought of as the thoughts going through their head.  Consider the scene itself:
Tom Robinson is transferred to the jailhouse in Maycomb.  Atticus sits outside the jail to keep watch.  Scout sneaks out with Dill and Jem to see what their father is doing.  They find him outside the jail and approach.  An angry mob of men also approach the jail.  Scout searches the crowd for a face she knows.  She spots Mr. Cunningham, the father of a boy she goes to school with.
Scout speaks to Mr. Cunningham.  She reminds him of who she is and that she goes to school with his son, Walter.  She tells him what a good boy Walter was.  She also refers to Mr. Cunningham's legal entailment.  After awhile, "Mr. Cunningham was moved to a faint nod" of recognition (To Kill a Mockingbird, chapter 15).  Mr. Cunningham softens.  He lets go of his anger long enough to bend down to Scout's level and speak to her.  He promises to tell Walter she said "hey."  He then stands up and tells the mob to disperse.
Mr. Cunningham saw Scout as a person who cared about his son.  He saw her as the daughter of Atticus.  He saw her as a human being.  His heart softened and he ordered the mob to leave.  When writing an internal monologue, you would focus on the change Mr. Cunningham experienced.  He let go of his anger toward Tom Robinson.  He realized that he did not want to hurt Scout or anyone in her family.  An internal monologue would show his inner voice changing.