In The Secret Life of Bees, Lily betrayed her mother when she killed her. What does this teach about the relationship she had with her mother?
We need to remember that Lily killed her mother as a child by accident. This was no deliberate act of violence in any way. Therefore I have to disagree with your question that seems to suggest that there is something to learn about their relationship because of the fact that Lily killed her mother. Actually, if we look at the text carefully, we can see that Lily's mother clearly loved her daughter very much. Even though she had effectively abandoned her daughter, this is something that can be shown in so many ways.
The problem that the text confronts is not the relationship between Lily and Deborah, but the way in which Lily has to work so hard to overcome her guilt and feelings of being "unlovable" because of what she did when she was a little girl. What Lily has to learn about her relationship with her mother is that she was and is deeply loved. Note how August has to tell her this:
But you're not unlovable. Even if you did accidentally kill her, you are still, the most dear, most lovable girl I know.
The true lesson to learn from Lily's killing of her mother is that no matter what we do we can still receive love. Lily killed her mother as a child when she did not know what she is doing. The novel shows us that there is always hope for a future no matter what happened in our past.