They're called 'icon boosters'.
There are a few cards that have a large struggle icon (identical to those on the story cards) printed in their text box. When such a card is either attached to a story or committed to a story, it forces an additional icon struggle of that type while the story is being resolved. This additional icon struggle is resolved according to the order of the printed icons.
Example: If Sleep of Reason is attached to a story, a Terror struggle would resolve four times–once as normal and one additional time immediately thereafter (before Combat) for each Terror icon brought by Sleep of Reason.
(2.27) Icon Boosters
Icon boosters are large struggle icons that add an additional struggle of the same type. Story cards with booster icons (either added by characters or support cards) are treated as having extra icons of that struggle.
Thus, if one icon was removed from the story card by a card effect, there would be one fewer struggle.
For example, if a player commits to a story that has Sleep of Reason (Core Set F53) attached (which adds three Terror struggles) and his opponent plays Inside Information (Core Set F151) to remove a Terror struggle, there would still be three Terror struggles to be resolved.
Icon boosters are always resolved immediately following the first struggle of the matching type (i.e., if the booster icon was Terror, there would be two Terror struggles before moving on to the Combat struggle) or if there is no struggle of the appropriate type it is resolved in the following order [Terror] [Combat] [Arcane] [Investigation].
Edited by jasonconlon, 28 October 2013 - 12:38 AM.