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The party Evoker has just reached 10th level and gained Empowered Evocation

Beginning at 10th level, you can add your Intelligence modifier to one damage roll of any wizard evocation spell you cast.

While most of his evocation spells do their damage on the round they are cast, only, he does currently know both acid arrow and arcane hand, which potentially have damage rolled over multiple rounds.

Acid arrow damages on the evoker's turn, regardless of whether it hits or not, and then damages on the target's next turn, if it hit.

Arcane hand lasts for up to 10 rounds if concentration is maintained, and each round one of five different effects may be used as a bonus action. Two of these five effects (clenched fist and grasping hand) deal damage on a hit, so there are potentially up to ten different damage rolls with this spell.

My reading of the ability is that so long as the bonus damage is restricted to a single damage roll, the caster can defer that bonus until a later round if desired - it is still a damage roll from an evocation spell that the evoker cast. I don't expect the player will ever want to wait until the second round to do damage with acid arrow, but because the arcane hand can be used on several different opponents over its duration I can see a tactical situation wherein the caster waits to a later round to apply the bonus damage.

However, I could also see an interpretation of "any wizard evocation spell you cast" implying that the damage bonus, if it is to happen, must happen on the round in which the spell is cast, not later.

For spells that damage over multiple rounds, can the evoker choose when to apply the extra damage from empowered evocation?

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You've mentioned the most important part:

Beginning at 10th level, you can add your Intelligence modifier to one damage roll of any wizard evocation spell you cast.

It does not stipulate when, only that it can only happen once per casting.

So for the acid arrow, it can either be on the initial damage, or if successful, on the follow up damage.

For the hand, the character can use the hand to block the first round and then smash on the second with the smash getting the bonus damage.

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