Translation:He has a headache and he needs to rest.
休息 simply means to rest, regardless the context. 轻松 can be used as a verb but more often is used as an adjective. (the characters literally mean "light" and "loose"). Here are a couple of sentences to give you the feel: 我要回家休息 - I will go home and rest. 我的工作今天比较轻松 - My work was relatively relaxed today.
'He has a headache, he needs rest' was marked wrong. The English is correct and it is a correct translation of the putonghua, but it was marked wrong. Once again, English has multiple ways of saying something that can perhaps only be said one way in Mandarin, all of these variations should be accepted as correct.