What noun or adjective describes a message/song/poem that is mournful, but intense?
In literature there is the term elegy (noun) or elegiac (adjective):
a sad poem or song : a poem or song that expresses sorrow for someone who is dead (M-W)
Yet it can be used by extension to describe 'woeful' situations without someone having died:
elegiac: wistfully mournful.
- She watched repeat serials, fixed on their moody and elegiac characterization.(OxfordL)
Edit: After your comment I thought of the adjective inconsolable which has some dynamic to it since whatever the person does, they cannot be consoled.
Harrowing is more extreme:
extremely upsetting because connected with suffering:
- a harrowing story (Cambridge)
For something more intense than mournful, perhaps:
After thinking about [some type of injustice] John moved those present with a heartrending song.
heartrending (adj.)
Causing great sadness or distress. Lexico
Causing intense sorrow or emotional anguish; evoking deep compassion; deeply distressing or moving. OED
Heartbreaking m-w