As a rule, yes. People despair for reasons, not because they're demented, for the most part. When you speak to someone struggling with suicidal ideation, that's what they say, as well. Bad things have happened that they feel aren't possible to recover from.
The word injustice doesn't encapsulate the entirety of the issue. Injustice happens, and people suffer as a result. Injustice alone doesn't create despair. Societal inequities, corrupt government and institutions, and cruel people in positions of power turn injustice into despair by making recovery impossible, or ensuring that depredations are continued repeatedly.
When people are trapped in situations that are intolerable, then they despair.