This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / Horatio Hornblower
The last paragraph of Hornblower and the Atropos. After having his ship snatched out from under him and getting sent back to England, Hornblower consoles himself that he'll at least be able to spend some time with his family... and he soon discovers that both his children are dying of smallpox.
There is a scene in Ship of the Line where the Sutherland comes upon a column of Napoleon's troops marching on a coastal road. For a few paragraphs, the point-of-view switches to the troops who are mostly conscipts. They quite welcome the sight of a pretty ship as a distraction on their long, hot march and even start waving at it. And then Hornblower opens fire on them. War Is Hell, indeed.
The death of Lieutenant Mound near the very end of The Commodore is a real gut punch. Likable, as clever as young Hornblower, and an excellent officer. Hornblower blames himself keenly for not having recalled the mission just five minutes earlier. It's one of the only times Bush is visibly emotional at a death.
Hornblower TV series:
The scene at Clayton's deathbed from "The Even Chance" is heart-wrenching. Clayton's sorry he's dying so very young and feels shamed by Horatio who was more courageous when dealing with their nemesis. He's sorry he didn't kill Simpson.
"The Duchess and the Devil": Archie's being convinced: "You don't need me." Fortunately also a Heartwarming Moment as Horatio insists that he does.
"The Wrong War"/"The Frogs and the Lobsters": Their relationship may be a Romantic Plot Tumour but Horatio's reaction to Mariette's death makes it hard not to want to hug him.
"Mutiny": Poor young Mr. Wellard, beating beaten up constantly because of the crazy captain. As Archie put it, it is injustice. Poor lamb. Break the Cutie indeed.
"Retribution": The rest of the movie after the battle aboard the Renown, complete with some in-universe Mood Whiplash. The audience saw Archie get shot but we never knew how serious the wound would be, and Horatio didn't see it at all. As soon as the battle is over, he and Archie are chatting, poking a bit of fun at Buckland... then Horatio looks over at his friend, notices that Archie is not quite all right, asks "Is that your blood?" and it goes downhill from there. From the courtroom scene in which Archie falsely confesses to mutiny, saving Horatio's from hanging, through Archie's death with his tearful smile and sad, sad goodbye, to Horatio's obvious devastation at losing his closest friend. The last fifteen minutes or so of "Retribution" are one big sob-fest.
Much of Horatio's interaction with Maria in "Loyalty" and "Duty". He's so obviously not as interested in her as she is in him, and even when he marries her his affection seems much more like sympathy or pity and a sense of obligation than anything else. You've really got to feel sorry for the poor girl.
When he is about to hand Betsy over to the Americans, without her husband, Napoleon's brother, who is to be returned to the French, Betsy tells him that she pities his wife. Hornblower replies simply that he does too, but that he is the man she married. He realizes what he is doing is a dick move of epic proportions, but his sense of duty means that he sees no possibility but to carry out his orders.