A 'wizard's lament fanfiction' might revolve around the wizard's regret over certain actions. For instance, if he made a deal with a dark force and now has to face the consequences. It could also be about his loneliness, as wizards are often depicted as solitary beings, and this fanfic delves deeper into his feelings of isolation.
Well, the central character is surely the wizard in distress. However, we might also see antagonists in the form of dark wizards or magical creatures that have caused his lament. There could also be a mentor figure from his past who appears in his memories as he laments.
When I write novels, I say that I write with emotion, which usually refers to the exclamation and reflection of a certain thing or situation. In a novel, emotional lamentation is usually a kind of emotional expression. By describing the protagonist's emotional changes and feelings about a certain thing or situation, the reader can feel the protagonist's inner emotions and inspire the reader to think. Sighing was an important way of expressing emotions in a novel. It could make the readers resonate with the novel and understand the theme and emotions of the novel more deeply.
The 'Lament of the Four Moons' in Asimov's science fiction could be a complex narrative. It may explore the relationship between the inhabitants of the world and the four moons. Maybe the moons are a source of power, and something has gone wrong with that power source, leading to a lament. It could also be a lament for the beauty and mystery that the moons once represented, but which has been lost due to some scientific or technological development in the story.
I can't confirm the name of the novel you mentioned because I don't have any information about it. If you can provide more details such as the genre of the novel, author, etc., I will try my best to provide you with a more accurate answer.