Paul Austen’s novel sold immediately to the author’s eager readers.
In the above sentence, which part is the complement and which is the adjunct? I am confused as to whether the adjunct should be immediately or eager readers.
"Eager readers" is simply the object of the prepositional phrase "to the author's eager readers." I would say that both that phrase, and the adverb "immediately" are adjuncts. Neither is necessary for the sentence to be grammatically complete.