“弄伤了背的”可以表达为“someone who has injured their back” 或者更简洁口语化的“back-injured (person)” （不过后者相对不太常见，更多在特定语境使用，前一种更通用自然）。
例如：
The person who has injured their back can't lift heavy things.（弄伤了背的人不能提重物。）
We need to be careful when helping the back-injured colleague.（我们帮助那个背受伤的同事时要小心 。 ）