To "let it slide" means to ignore it (or not say anything about it) when it shows up. Without context, I don't know what "brokenness" the text is talking about. The narrator says nothing about that "brokenness", because he does not want to hurt her pride, but at the same time he admits he may have hurt her pride in order to spare his own pride. Let me know if this is still unclear!

Hello Pei Yung, The original essay was written by a young lady and her experiences of learning English and teaching her mother. She is saying that sometimes she didn't point our her mother's mistakes to spare her mother's pride but maybe doing this caused her mother more problems. Hope this helps Bob

In context of the paragraph from which you have highlighted this sentence, The author is simply referring to the action she takes while teaching her mother to read and speak English. In the previous sentence, the author mentions that she feels guilty when sewing her mother's words together. This most likely refers to the sense of discomfort when correcting her mother's usage of words in the English language. Then we come to the main sentence "Sometimes I let the brokenness slide to spare her pride” this means sometimes I overlook the errors in her words to not make her feel ashamed. "But perhaps I have hurt her more to spare mine" meaning I point her errors more than I ignore it so I do not feel ashamed of my mother's use of the English language. So put it together, we can say that the Author teaches her mother how to use the English language. During her teaching sessions, she sees errors in her mother use of words. Although she ignores some of her mother's errors so as to not make her feel ashamed, she points out more of these errors because she is more afraid of feeling ashamed of her mother's English literacy than her mother feeling ashamed