The phrase 'lessen the damage' is correct and usable in written English.
You can use this phrase when you want to talk about reducing or minimizing the effects of something bad or harmful. For example, "We need to implement some strategies to lessen the damage caused by the flooding."
A more accurate term is "mitigating risk" – taking measures that will lessen the damage and loss of life in future events.
After an intense debate, the assembled countries resolved to search for ways to lessen the damage of what they called increasing rates of emigration.
The question of what can be done to lessen the damage is vexing and gets into difficult questions of whether to limit the migration of skilled workers.
What if the power company could have sent in robots early in the crisis to lessen the damage, and now to clean up the irradiated dead zone?
We're used to hearing all kinds of advice intended to help us battle against the tide of technological mishaps (or at least, lessen the damage when they occur).
Irisin, a hormone that is released during exercise, may improve brain health and lessen the damage that occurs during Alzheimer's disease.
The neighbors of the municipality worked fast and efficient in an effort to lessen the damages.
It can effectively lessen the damages to the protein membrane at the top layer of our skin caused by over-exposure to sunlight.
If you are the non-breach party in a contract, you can seek to mitigate (lessen) the damages caused by the other party's breach of contract by seeking replacement goods or services (known as "cover").
Southend lessened the damage with a late Nicky Bailey goal but by then it had ceased to matter.
While the Mozambican government's flood plan, announced last year, may have lessened the damage, Malawi was caught off guard.
