Who is the boy named "Cunningham" in To Kill a Mockingbird? I believe his name starts with a 'y'.

The young Cunningham who appears in the novel is Walter Cunningham, a classmate of Scout's.  He is the son of the older Walter Cunningham, a man for whom Atticus is serving as lawyer.  The Cunninghams are poor, but proud, and Scout takes it upon herself to explain this fact to their new teacher, Miss Caroline.

You mentioned a young character whose name starts with a "y".  There are no characters that match that description, but perhaps you were thinking of Burris Ewell (the last name is pronounced YOO-wuhl, hence the "y" sound).  Burris is another of Scout's classmates -- at least for the first day of class -- and is also poor.  Unlike Walter, Burris belongs to a less proud class of poor.  It is his father, Bob, and his sister, Mayella, who will factor heavily into the trial portion of the novel.