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The Draconic Bloodline sorcerer feature Elemental Affinity states:

[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 dragon's breath spell (XGtE, p. 154) incorporates a damage roll, but it is not a direct consequence of the spell:

Until the spell ends, the creature can use an action to exhale energy of the chosen type in a 15-foot cone. Each creature in that area must make a Dexterity saving throw, taking 3d6 damage of the chosen type on a failed save, or half as much damage on a successful one.

Would one of these exhalations qualify for the application of the Elemental Affinity feature? If so, is only one eligible or every one throughout the duration of dragon's breath

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You can use it. Once.

It's a spell, and it does deal damage of the chosen type (assuming you choose the same type for the dragon's breath as you have for your bloodline). That meets the requirements for applying the class feature to that casting of the spell. The spell likewise has associated damage rolls, and thus the bonus can be applied. There is no listed requirement that anything be a "direct consequence" of anything. It's just as applicable here as it would be with wall of fire or produce flame - each of which applies damage at times and based on triggers other than the instant of the casting.

This, as stated in the class feature, allows you to add your charisma modifier to one damage roll of the spell - in this case one exhalation. Any other exhalation would be a different damage roll, and wouldn't get the benefit. Timing here could get kind of odd, amusingly, as it never tells you you have to add it to the first damage roll of the spell, and it doesn't tell you if you decide whether or to add it before or after you roll the dice, but the "Yes it works, use it once." part is pretty clear.

The "one damage roll per spell" is accurate, though was not in the original printing. It's an errata.

Per The Player's Handbook Errata,

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

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    \$\begingroup\$ Citations are needed to support this answer. \$\endgroup\$ – NautArch Mar 19 at 1:01
  • \$\begingroup\$ @NautArch it's a read straight off of the quoted text. It's really pretty straightforward. It's just like the same question with Produce flame. \$\endgroup\$ – Ben Barden Mar 19 at 13:11
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    \$\begingroup\$ Your alternate interpretation does not seem to follow how area of effects work in the game. See the below questions for details: Is Thunderwave’s damage rolled once for all targets or rolled for each? \$\endgroup\$ – David Coffron Mar 19 at 13:45
  • \$\begingroup\$ @DavidCoffron Excellent! thank you. Fixed. \$\endgroup\$ – Ben Barden Mar 19 at 14:04
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    \$\begingroup\$ @Molot there was an errata. I've added a pertinent reference to my answer. \$\endgroup\$ – Ben Barden Mar 19 at 15:31
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It does not apply, the damage is caused by spending an action - not casting a spell.

As quoted above, Elemental Affinity engages when(emphasis mine):

you cast a spell that deals damage of the type associated with your draconic ancestry...

When you cast dragon's breath, you are not dealing damage. You are granting an ability for whomever it is cast on to(emphasis mine):

use an action to exhale energy of the chosen type in a 15-foot cone.

It is when you use that action, not when you cast the spell, that is dealing the damage.

You are using your action to utilize the effect granted by the spell, but not spell is being cast. The spell is only giving the creature the ability exhale energy.

**You are not meeting the requirement of casting a spell when you are dealing the damage and therefore can not use your Elemental Affinity.

The damage dealing effect is similar to Breath Weapon of a dragonborn.

There is no spell being cast, it is just an ability that uses an action to engage. When you deal the damage, you are expending an action. Granted, it's an action given to you from a spell, but the damage is dealt from the action - not from casting a spell.

The fact that a spell gave you that ability doesn't change the mechanic of its use. When you're dealing the damage, it is not from casting the spell. It is from using the action granted by the spell (which, again, is not casting a spell.)

Strict vs loose reading

Ultimately, I think there is some ambiguity and it depends entirely on how you read the spell. Given that we are talking about a feat here and one concentration-based spell, I don't think it'd be a huge deal to grant it by stating that the spell is ultimately causing the damage on one breath attack action during the spell duration.

This is somewhat supported by the same initial line. The difference is in granting

cast a spell that deals damage...

to the end-action for the damaging effect and not just the spell itself.

If it becomes overused, then you can always go back to the stricter after discussing it with your player.

But the hard line reading does suggest that this interaction doesn't work given you aren't casting a spell to cause the damage.

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    \$\begingroup\$ @BenBarden You've got your answer to counter mine. Let's not debate in comments. I've attempted to address my logic regarding what the spell is doing/granting and why it's different than what Elemental Affinity works with. \$\endgroup\$ – NautArch Mar 19 at 14:22
  • \$\begingroup\$ Fair. I think your logic hinges on you using a stricter reading of the word "when" than I do. I read it as being part of the conditional. You seem to be reading it as requiring that the benefit be claimed int eh same instant as the spell is cast. If that's true, your answer might be improved by calling that out explicitly. \$\endgroup\$ – Ben Barden Mar 19 at 14:25
  • \$\begingroup\$ Yay. Now we have both answers that I couldn't figure out which is better... Still not sold on either argument... Guess I'll wait to see where the votes land \$\endgroup\$ – David Coffron Mar 19 at 14:43
  • \$\begingroup\$ @DavidCoffron Totally up to you, but I tried to support mine with the reading of the rules text and similar abilities :) It's definitely not the answer you probably want - but I'm having trouble seeing the flaw in my argument(unless it's just that i'm being too strict with the wording.) Let me know if you think i can improve it or if I've missed something, please. \$\endgroup\$ – NautArch Mar 19 at 14:44
  • \$\begingroup\$ @NautArch The logic could go both ways in my eyes. Either the spell must directly cause the damage, or the spell must result in causing damage. It is ambiguous fmpov. shrug \$\endgroup\$ – David Coffron Mar 19 at 14:48

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