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According to the PHB Errata:

Elemental Affinity (p. 102). The damage bonus applies to one damage roll of a spell, not multiple rolls.

And there's the ice knife spell (XGE, p. 157):

You create a shard of ice and fling it at one creature within range. Make a ranged spell attack against the target. On a hit, the target takes 1d10 piercing damage. Hit or miss, the shard then explodes. The target and each creature within 5 feet of it must succeed on a Dexterity saving throw or take 2d6 cold damage.

If a Draconic Bloodline sorcerer with White Dragon Ancestry casts ice knife, how does Elemental Affinity's damage bonus work?

Does the damage bonus apply only to the explosion afterwards? Do all enemies caught by the cold damage take the additional damage of the Charisma modifier?

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You pick which (one) damage roll it applies to, but the wording suggests it'd apply to the "elemental" cold damage

The Draconic Bloodline sorcerer's Elemental Affinity feature says (post-errata):

Starting at 6th level, when you cast a spell that deals damage of the type associated with your draconic ancestry, you can add your Charisma modifier to one damage roll of that spell.

The feature doesn't say how the damage roll it's applied to is chosen, so it's up to the caster. That said, the way Elemental Affinity is phrased suggests that it's meant to provide a boost to the damage of that element type.

The ice knife spell deals only (1d10) piercing damage on initial hit. It's only the subsequent explosion of ice that deals the (2d6) cold damage. As such, it seems likely that the feature is intended to boost that cold damage, given that the feature only activates because you're dealing the cold damage. If applied to the cold damage, then, well, the damage bonus applies to that roll as a whole - so it'd increase the damage dealt to everyone within the area of effect that failed the Dex save.

If, however, you cast a spell that dealt the damage type associated with your dragon ancestor on multiple instances - such as a concentration spell like call lightning, assuming your dragon ancestor is blue/bronze (associated with lightning damage) - then you'd get to choose which single damage roll you'd add your Charisma mod to. You wouldn't get the Charisma mod bonus every time you dealt damage with that casting of the spell; you'd get it just once per casting, but you could choose which damage roll to boost.

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To address your last question, the relevant rule from the Basic Rules is under “Damage Rolls”, in the “Damage and Healing” section of Chapter 9: Combat:

If a spell or other effect deals damage to more than one target at the same time, roll the damage once for all of them. For example, when a wizard casts fireball or a cleric casts flame strike, the spell's damage is rolled once for all creatures caught in the blast.

So in the case of ice knife, if you choose to apply the bonus to the 2d6 cold damage roll, the extra damage will apply to all creatures in range.

Note that if the creature has resistance, you add the modifier to the roll before halving the damage. (This also applies for spells where a successful save results in half damage, though ice knife is not one of those; creatures who save take none of the cold damage.)

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