"After he heard the news, he cried sadly."
It is barely alright, because you have left out 這個, the clauses still maintain a close correlation. If you add back 這個 and some other details, the sentence will begin to fall apart. You would need some "glue" to relate the clauses tightly e.g. 他听了这个令人震惊的新闻之后，难过地哭了。
The sentence syntax in Chinese is like a tree. You draw the stem first and then the branches, and then the leaves. Here the "stem" is 他哭了. 听了这个新闻 is a "branch", describing in which circumstance the stem meaning happens. It is why the 他 in this part is omitted rather than the latter one.