When you read the series, the state of despair and hope is almost like a cycle. I know that @gp often likes to emphasize on what state of hopelessness Narnia is in yet in the end everything turns out alright (shout out to @gp if you are reading this).

Here how it's played out chronologically-

The Magician's Nephew- Digory is in a hopeless situation. His mother is sick and is about to die, his father is away in India. Now the world of Narnia had been newly created. However, it's already been marred by evil. So it may have seemed like a despair, but Aslan says, "I will see to it that the worst of it falls on myself" (He was foreshadowing His own death on the Stone Table in The Lion, The Witch, And The Wardrobe). So He was reassuring that even though the world of Narnia has been marred by evil, it's going to turn out alright in the end. Aslan even rewards Digory for his obedience to bring back the silver apple and planting a Tree of Protection by giving him a silver apple. Digory takes it back to our world, and uses it that heal his dying mother.

The Lion, The Witch, And The Wardrobe- The world of Narnia is in a state of hopelessness. The eternal winter is despair. When the White Witch took Christmas out, the "always winter and never Christmas" is a picture of a world without joy and hope. But the hope is that prophecy is about to be fulfilled when the Pevensies arrive, Aslan has returned to Narnia, and it's the return of spring. Christmas has to come first before spring could come. There is a despair when Aslan dies on the Stone Table, then there's hope that He came back to life. Shortly after, Aslan frees the statues at the White Witch's castle, and not long after, the prophecy is fulfilled.

The Horse And His Boy- Now this is a book that takes place during The Lion, The Witch, And The Wardrobe. It's a hopeless situation for Shasta and Aravis. For Shasta, he's been abused by a fisherman Arsheesh whom he calls father though not really his father so constantly. And he's about to be sold as a slave to Anradin Tarkaan, who is the original owner of Bree. For Aravis, she is about to be married of to a much older guy, Ahoshta at the suggestion of her stepmother who actually hated her (I know, a lot of stories about stepmothers who are not fond of their stepdaughters). At the time they get to Archenland, Shasta realizes that he was meant to save Narnia and Archenland from Calormene invasion. Spoilers: he's actually Prince Cor of Archenland. So it's like he rose to a higher power, making him realize that this is what he was meant to do. Aravis learn so much about humility, and even gets a lesson from Aslan Himself who scratch her back (ouch!) She needed to know what it felt like for her stepmother's maid who was most likely beaten. Yet she learns that there is so much more to life.

Prince Caspian- Narnia is in state of hopelessness again. Telmarines have taken over, the old days are gone, and even Old Narnians living in hiding began to doubt that Aslan was even real. Caspian X, with the help of the Pevensies, bring back the old days to help change the path of Narnia's history. Aslan even tells Lucy, "Now Narnia will be renewed." When Aslan makes a roar, the dryads and naiads wake up from their long sleep, and frees the school children. There's also a moment where Caspian and his nurse are reunited after years of separation.

The Voyage of the Dawn Treader- There isn't a lot about despair and hope in this book. The darkest moment is The Dark Island, where "worst nightmares come true". This unseen world with total darkness where dreams become reality. It's a hopeless situation to be in. However, Lucy calls out to Aslan, and Aslan answers her by coming in form of a albatross, leading them to the light. He even destroys the Dark Island once and for all. So out of darkness into the light.

The Silver Chair- There's a hopeless situation for Eustace and Jill at school. There are bullies, and there's nothing anyone will do about it. When they get to Narnia, Narnia is in state of despair. Caspian was about to die without an heir, and they've already lost many of their bravest Narnians trying to find Prince Rilian. There is hope when Prince Rilian is back in Narnia and reunited with his father for a brief moment. Caspian dies, and even Aslan mourns for him. Soon after, Aslan instructs Eustace to pluck a thorn, drive it into Aslan's paw, and Aslan use's his blood bring Caspian back to life.

The Last Battle- Narnia is in a state of despair once again. Shift is taking over Narnia by conspiring with the Calormenes, claiming that this is the "will of Aslan", Calormenes are taking over Narnia as they have been wanting to for a long time, Tirian is tied up to a tree, and Cair Paravel is ransacked by the Calormene navy. However, as Tirian is tied up to the tree, he holds on to the hope by remembering what had happened in Narnia's past- "When things were at their worst, it came out alright in the end." Though Aslan brings an end to the world of Narnia, the hope is that all of those from Narnia's past (and there's a whole list of them) are together again in the new Narnia. So it shows that death is not the end of the story, but a new beginning.