It’s a rare sigh that does not have matter in it. This moment is interesting in that Gertrude is clearly sighing, clearly upset and her husband doesn’t say, “What’s the matter?” “What happened?” or “What can I do?”
He doesn’t comfort her at all. He just states the obvious and tells her what to do.
There’s a question of whether there was or is genuine love or even lust between these two characters and I wonder if that’s a question for Gertrude, as well. When she begins this scene sighing, is she hoping Claudius will comfort her, ease her distress and/or fears?