What impact does the absent father have on the characters and the story as a whole?
The figure of the absent father was probably best explored for its literary signficance by Mark Twain in his most famous works The Adventures of Huckleberry Finn and Tom Sawyer.
Ernest Hemingway also offers a wealth of commentary on this situation in his work, especially in For Whom the Bell Tolls where the father figure has committed suicide.
Each of these writers look at the son's sense of honor and identity as being impacted by the father's absence. The son encounters a set of choices which are understood to be choices that would not have been presented to the son if the father had been around.
This leads to a crisis of identity but also an urgency in moving into adulthood.
The father's absence does not equate to a lack of importance for his (empty) role. On the contrary, the story only becomes what it is in the cases cited here as a result of the father being absent.
Other works that present character with absent fathers: The Life & Times of Michael K.: The Lord of the Rings: A Raisin in the Sun.