Hugh, Ulf, Addam, and Alys: What I think the point of that scene in S03E01 was
In the show, Hugh, Ulf, and Addam are sent to the banks of the Gods Eye to lay in wait for Aemond. Ulf has a weird poop and they all get spooked back home by Alys. I've seen some people say this was a pointless change from the book, but I think there are 2 or 3 key story points from the book that the scene was meant cover in addition to the realities of TV production.
Spoilers abound for later this season and probably the first half of Season 4.
Point #1: In Fire & Blood, we get the following after the Battle of the Gullet:
> Mushroom tells us there were two men on Dragonstone that night who drank to the slaughter in a smoky tavern beneath the castle: the dragonriders Hugh the Hammer and Ulf the White, who had flown Vermithor and Silverwing into battle and lived to boast of it. “We are knights now, truly,” Hard Hugh declared. And Ulf laughed and said, “Fie on that. We should be lords.”
> The girl Nettles did not share their celebrations. She had flown with the others, fought as bravely, burned and killed as they had, but her face was black with smoke and streaked with tears when she returned to Dragonstone. And Addam Velaryon, lately Addam of Hull, sought out the Sea Snake after the battle; what they spoke to each other even Mushroom does not say.
We get Nettles/Rhaena being traumatized by the battle and Addam seeking Corlys in Episdoe 2, but not Hugh and Ulf's celebration. Instead, the showrunners moved that key bit of characterization to before the battle: Ulf starting to realize that the all the Blacks are offering him is a knightship and in return for taming a dragon larger than any the Blacks have besides Vermithor.
Here's where I'm going to sneak in the production realities reason for the scene: having the three of them hang out by a lake is both cheaper than having six dragons at the Gullet all at one time and lets the drama of battle play out with less chaos for the viewer. Alys pops up to tell them they're late to the battle which gets them back on their book path and lets them get battle experience. Production reasons are a lot less fun to talk about than story issues, so I'll leave that there.
Point #2: Much later on in Fire & Blood, we get the following:
> The dragon was Seasmoke, his rider Ser Addam Velaryon, determined to prove that not all bastards need be turncloaks. How better to do that than by retaking Tumbleton from the Two Betrayers, whose treason had stained him? Singers say Ser Addam had flown from King’s Landing to the Gods Eye, where he landed on the sacred Isle of Faces and took counsel with the Green Men. The scholar must confine himself to known fact, and what we know is that Ser Addam flew far and fast, descending on castles great and small whose lords were loyal to the queen, to piece together an army.
Addam is rumored to have visited the Isle of Faces between when he leaves King's Landing and when he shows up to start the Second Battle of Tumbleton. Before that point in the book, there's no connection between him and the Gods Eye that I can see but this has fascinated fans for ages. To my knowledge, Addam and Howland Reed are the only characters in the entire series said to have visited that island so this is kind of a big deal.
Point #3: around the same time as the last point, we have this paragraph:
> For all these reasons, Lord Hammer (as he now styled himself) began to dream of crowns. “Why be a lord when you can be a king?” he told the men who began to gather round him. And talk was heard in camp of a prophecy of ancient days that said, “When the hammer shall fall upon the dragon, a new king shall arise, and none shall stand before him.” Whence came these words remains a mystery (not from Hammer himself, who could neither read nor write), but within a few days every man at Tumbleton had heard them.
This prophecy eventually finds its way to Hugh and to men who start to flock to him. This is my least strong point, which is why I kept it for last. The show deals a lot with prophecy, but there's only one prophecy in the book, this one. As you can see, the talk of prophecy kind of comes out of nowhere with no source ever being identified.
TL:DR: I believe that scene sets three things from the book up:
- Ulf realizing how little reward the Blacks plan to give him despite him having one of the largest dragons
- Addam being introduced to the Green Men
- Hugh being introduced to one of the major sources of prophecy in the books, greenseers
As to whether the scene works, that's an exercise I leave to the reader. My point is just that I believe I see what the idea behind the scene was.