I present to you a very much needed compilation of Henry going:
Here's a breakdown of every "mh" (cause I have no self control over how many times I've watched this movie), and the different feelings they give off :D
Alex iconic "Vicky... classic." To which Henry fakes a smile and does "mh", trying to sound convincing on their genuine friendship, while probably also questioning himself why must've been him the one he fell for.
The conversation about owning a key. The "mh" is bittersweet and reflects his need to cope with humor on what it's required from him as prince and all the things he can't do because of it.
"Have you ever got your heart broken?" "The day my dad died." It's a sad freaking "mh". It holds melancholy, longing and grief but again, it helps him cope with the situation.
Honest, spontaneous, interested "mh" for whatever Alex was about to say and thinking because Henry is always ready to hear all the intrigued thoughts that come out of Alex's mouth.
Lastly, an alerted, concerned "mh" for whoever made his boyfriend so uneasy and he's ready to comfort him and give him all the reassurance he needs.
Feel free to give your own interpretations and thoughts