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The phrase "already received compensation" is correct and usable in written English.
It can be used when indicating that someone has been compensated for something prior to the current moment or context.
Example: "Since you have already received compensation for the damages, we cannot process any further claims."
Alternatives: "previously compensated" or "compensation has been provided."
Exact(3)
But, crucially, customers who have already received compensation for PPI mis-selling will not be able to claim again.
Some of the victims of the blood scandal who had evidence of the batch numbers and could clearly identify which products caused their infections have already received compensation from pharmaceutical companies, he added.
The family of a victim who was forced into slave labor may seek restitution even if the family has already received compensation from the German government at least under the rules of the international commission.
Similar(57)
Critics contend that it makes no sense to try to redress wrongs that reach back through the centuries, and that Caribbean countries already receive compensation through development aid.
The family already receives compensation for tweeting ads to millions of followers and making public appearances at club openings and other events.
The $72.7 million would be in addition to $120 million investors have already received in compensation, most of it from bank accounts of the founder of the chain, Eddie Antar.
He has not yet received compensation, though.
You have now received compensation of £596.33.
Steve received compensation in 2009 – £40,000.
In 2000, he received compensation of $5.2 million.
About 35% of petitioners have received compensation.
More suggestions(3)
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com