Here, without context, use the present perfect particularly for exams. (... the letter has been sent.) In casual speech, Americans and Canadians often prefer the simple past. (... the letter was sent.) Here is a brief overview of when to use the present perfect and the simple past. https://www.ef.com/ca/english-resources/english-grammar/present-perfect-vs-simple-past/

The first ones are very direct and work with a yes or no. Yes, the letter was already sent Yes, the grass was already cut The second work a bit further and more friendly/fluently. Who sent the letter? "The letter has already been sent" to the customer Why are they cutting the grass? "The grass has already been cut" this week