Attempting to keep things vague to avoid possible spoilers for Dragon Heist

During the first session, after the barfight, a couple monsters crawl out of the hole in the middle of the tavern, one big scary dude and three bugs. The 'Big Scary Dude' would be worth a ton of XP if killed by the players, but there are two caveats

1) It is at half health when introduced

2) The players are told in no uncertain terms to focus on the little bugs while an NPC takes care of it.

Before the players even have a chance at him, the NPC goes before them in the Initiative and outright kills the big scary dude with a four hit combo on his turn. The big dude goes down literally before they even have a chance to react.

Just the other day, one of my players was asking me if they should have gotten experience for being apart of that battle with the big dude. If they did, they would be undeniably level 2 for the rest of the first chapter, but my thoughts are that they did not actually do anything in that fight and therefore should not get the XP. Am I in the wrong for denying them the XP? Or am I in the right for maintaining continuity?

I personally feel Dragon Heist isn't optimized for using experience

Waterdeep: Dragon Heist seems to be designed with the milestone level system in mind, and it accordingly gives out roughly one level per chapter. If you've decided that players get experience for killing monsters, instead of levels for completing chapters, then you've already set yourself up for more problems down the road, because Dragon Heist does not have nearly enough combat encounters to get you to the required level for the later parts and you'll be forced to improvise extra experience for social encounters and traps.

So as far as the reasoning is concerned, no, they get no exp for it, because they never get exp for anything, they get a level when they complete a milestone.

But I've already decided to use experience!

Even then, they still shouldn't really get any experience for the big monster, because they weren't involved in fighting it. The NPC was in a fight with the giant monster, they were in a fight with the small things around him. Nothing they did contributed in any way to the fight with the big monster, and nothing the NPC did contributed to the fight with the little monsters.

For all intents and purposes, they were two separate encounters, and one of the encounters did not involve the players, they were essentially bystanders. If your player insists that they deserve experience because they were also there for the encounter, then logically speaking all other NPCs in the tavern were also there and should also share in the experience.

Yes they should get XP for the encounter, but it should be divided between the NPC + Party

Dragon Heist gives you the option of either using XP based advancement, or using milestone advancement. If you choose to use XP based advancement (which you clearly have) then you should be dividing out XP for the encounter as a whole.

Based on your description I believe the encounter in question is the

Troll and 3 stirges

While your party did not actually fight the

Troll

due to lack of opportunity they did participate in the encounter. In particular they stopped

3 stirges from dogpiling on Durnan while Durnan fought the Troll.

Similarly,

Durnan

prevented the PCs from being attacked by the

Troll

Finally the module expects the PCs to be thanked by

Durnan

for their help in the encounter, and stabilised if they have been reduced to 0 HP.

One thing to remember is that the

Troll regenerates back if it isn't hit by fire when it's at 0 HP. Durnan has no way (as written) to deal that fire damage, so he needs help from somewhere to actually deal the fire damage. The module mentions Durnan dousing the Troll in lamp oil, but that should have at least taken one of his 4 attacks. If you allowed Durnan to throw the lamp oil and make four attacks then you changed the action economy of the fight.

As you mention, the

Troll

is at half it's hitpoints when the encounter starts. This does (technically) reduce it's defensive challenge rating, which will then have a knock on effect on it's overall challenge rating. I have the calculations for this at home and will dig them out (later). This does add additional complications to the calculation though.

You always have the option of not dealing with this complication, and just awarding the normal XP for the monster.

If we use the normal XP for a full health monster, the total XP for the encounter is:

1800 + 3 * 25 = 1875

If we assume you are playing with 4 party members then the encounter XP should be divided by 5 to get the per player XP.

The would mean each player should be awarded

375 XP for the encounter

If you don't include the

Troll

And treat them as two separate encounters, then they would get

18 XP each. (75/4 = 18.75 which is rounded down to 18)

But, they didn't actually hit it?!!

Lets take this argument to the extreme. If you have an encounter setup for your party which has 50 goblins walk around a corner. Your party has a Sorcerer (or a Wizard) with fireball, who happens to roll high enough in initiative that they go before everything else in the encounter (both goblins and other PCs).

The Sorcerer seizes the moment and casts Fireball, incinerating all 50 goblins.

Should the other PCs in your party get an XP share? or should it all go to the Sorcerer?

To put it another way, if your party has an especially low Dex PC who doesn't roll high enough to participate in the encounter (the rest of your party are efficient at their monster murdering). Should that low Dex PC be excluded from the XP calculation?

No. You get XP for overcoming threats, not for letting someone else do it.

Basic Rules:

As your character goes on adventures and overcomes challenges, he or she gains experience, represented by experience points.

The scenario supplied them with an NPC whose job is to fight the monster and keep it away from the PCs. The NPC did his job, thus preventing the party from ever having to engage with the monster at all. They were, as Buffy the Vampire Slayer would have put it, "fray-adjacent".

If Godzilla is going to eat your peaceful village, and a hero goes and drives Godzilla away, what you did was get rescued. You don't get a share of the XP for that. Your reward is to not get eaten.

The typical D&D party is very often in the position of the hero here: being invited or contracted to deal with someone else's problems. When the Lord of Castle Aaaargh hires you to slay a dragon for him, does he get XP for that? No! Even though, if you had failed, he might well have had to put on his armor and go out there himself. Because that's not what happened. He outsourced the job, so he outsourced the glory.

If you want to advance to hero status, you have to step up to Godzilla in some way. You don't have to kill him, or even fight him. You could decoy him away from the town, or dig a giant trench to stop him. But you have to act.