Destroy is the action used to break structures. Destroying structures drains drains Phlegm. Structures take % damage while being Destroyed, once the structure reaches 100% damage it will break. The % damage done is related to your remaining Phlegm and that structure's Soak Value, so your rate of destruction slows down as your Phlegm is drained. Attempting to Destroy Claimed stuctures is a Crime and will leave Scents and summonable Evidence of Waste.
Can Be Destroyed:
Most structures have a Soak Value which determines how much Phlegm a pilgrim needs in order to deal damage to that structure, having less than that value will cause you to do 0 damage to that structure when attempting to Destroy it. The Phlegm needed is roughly 2x the Soak Value. Some structures have such a high Soak Value that you will most likely be unable to Destroy them by regular means. In those cases Greek Fire can be used. Using Salvage, when it is possible, will ignore Soak Value.
Alternate Methods of Destroying
Certain stuctures require an alternate method to Destroy them due to the high investment in building or maintaining them.