It comes down to context. If the story is close third person and this is the PoV character, then "He approached cautiously" is fine, but it might be better to describe why the character felt the need to be cautious. If this is a non-PoV character, then I would suggest describing how the character approached to make it appear cautious. Of course, in either case, if cautiousness is not important to the situation, consider omitting the word.
The second example, "his face flooded in relief," gives me more pause. The PoV character cannot view his (or her) own face. In this case, it would be better to describe the internal thoughts that led to the feeling of relief. For a non-PoV character, "flooded in relief" is an interpretation of what is actually observed. Describe how the character's demeanor has changed. If, in either case, the additional words do not feel justified, consider whether the "flooded in relief" is really important to the scene.
For me, one of the lessons of "show, don't tell" is in deciding what is important and what should be omitted. If something is important, engage the reader with it. If it is unimportant, perhaps it is unnecessary.