Is it true that to borrow books by H.L. Mencken from the library, Richard Wright pretends that he is borrowing them for his master, a white man?

There is some truth to this.  In the thirteenth chapter of Black Boy, Richard Wright's autobiography, he describes his experience going to the library.  Segregation laws prevented African Americans from checking out books at the public library.  Richard Wright borrows the library card of a white coworker.  Wright then forges a note from his coworker, giving himself permission to use the card to check out books. 
At the library, the staff treats Wright rudely.  Eventually they allow Wright to check out the books by H.L. Mencken.  Wright reads the books, which change his life.  Mencken's writings also inspire Wright to become a writer himself.
While the situation described in your question is true, one of the circumstances is not.  Richard Wright was born in 1908, over forty years after the end of slavery in the United States.  Wright lived under Jim Crowe segregation, but he was never a slave.  This means that he did not have a white master.