This is first time DMing with two family members (who have had experience playing D&D), and a friend who has never played before. By luck of the draw, when we started up the game, we had a cleric, a wizard, and a rogue, but no fighter and none of their hit points were above 10hp. To make things easier and more fun for the first time player (our cleric), I provided a barbarian fighter NPC who would level up with them (so she's not overpowered) and only plays a role during violent encounters. My rogue, however, (a verging on chaotic evil, bugbear) has decided to take her existence as a personal insult, and a punishment, and has been passive-aggressively telling me (out of game) that he's got it under control and they don't need a fighter, nevermind that his character got one-shotted by a skeleton and when he was revived decided to abandon his team-he has explained to me that he had "planned" to just take a little bit to recover and than go back in, if the barbarian hadn't show up. He feels like the barbarian is cramping his style, despite the fact that she never interacts outside of violent encounters, doesn't say anything, and specifically plays no role in the plot of the game. I have no problem killing PCs, but maybe not at level one, with a one shot, because the only character with any fighting capabilities decided to run for it. I have discussed her role with this player, explaining that as soon as they make it out of this dungeon (around level 3), she will leave. As you can tell, I'm getting very frustrated with this situation and do not know how to resolve it; this player, throughout the various games we've played, consistently is a punk, and argumentative when things don't go the way he feels they should go. The other players like the barbarian and are glad she's around and have no issue with her existence. Have you had a similar situation and how did you resolve it?

Suggestion: The barbarian takes a one-encounter holiday so you can see whether the rogue's player is right.  Make sure said player knows that you're doing this. 

DnD doesn't require the MMO-Staple Tank/Healer/DPS setup. Honestly the cleric can be very similar in durability as a fighter. And you as the DM are entirely responsible for encounter design and execution. Creating encounters for a group of 3 is harder than a group of 4, but if you are running a homebrew campaign you can adjust them as needed.

As a DM you control both sides of the equation, you are adding 1 to the hero side to make them equal, when you can just as easily subtract 2 from the other side.

Thank you!

I'm using a pre-written dungeon from Tales from the Yawning Portal (Sunless Citadel) that will flow into one I'm designing at the dungeon's completion. I'll take a look at editing the encounter numbers, since they are starting to get into more difficult territory. Thank you!

There are a few ways I would solve this. 

1) As mentioned before, let the rouge try and handle it, or even the cleric (built right clerics can be brutal in any party role)

2) Ask the rogue to roll up a secondary character for him to run, since it sounds like they have a bit of an ego frame it as he has attracted a follower/bodyguard to protect him. 

3) Scale encounters down. Looking at the DMG has instructions for calculating your XP budget and how much XP adventures should be aiming for on an adventure day. 

4) Buff-em up. This is what I would go for personally, in the DMG there are rules for Boons/Charms/Gifts that give characters extra powers and abilities. These are normally limited in the number of uses or for how long they last. Thes can be given by items, magical areas, some deities and even monsters. Have your rogue impress a god of thieves and be awarded some extra hp for a while. 

Also as was mentioned before, D&D is not an MMO, you don't have Tank/DPS/Healer slots that have to be filled. Instead look at it more as what role does this character fill in the party? Is he a meat shield for soaking up damage? Is he a damage dealer that drops guys? Healing? Buff/Debuff? Support? Out of combat utility/skill monkey? As a DM it is not just your job to run the game but to make sure that the game is appropriate for your players.

In addition to the above points, I think it's fair to point out that every adventure in TftYP, including (and maybe especially) The Sunless Citadel, is designed to be a particularly difficult adventure... and that's for a standard 4-6 PC balanced party. If you're working with 3 players, and none of them are martial classes, there's no shame in allowing your PCs to start at 2nd level, even allowing them to be one level higher than the adventure recommends at any given time (I'd end this handicap at 4th level, though). This can allow for the players to enjoy the experience of the adventures' story while shielding their squishy selves from the errant critical hit from a goblin, kobold, twig blight, what-have-you. Personally, I'm less and less a fan of starting PCs at 1st level anyway... especially if most of them are experienced players. I find that the amount of latitude this gives me as a DM is huge, and it precludes every-damn-single-random-encounter from turning into a white-knuckle affair, both for them, and for me behind my screen. What's the point of picking up a nat 20 behind my screen because I don't want to start off my adventure with an insta-death of the party sorcerer?