And of course if she REALLY loved and understood him, she would already know what was bothering him.
AKA, the same crap women tell men when they are all upset and the guy’s supposed to read her mind.
I’m confused at the logistics of the TV hug. Such a hug would come from behind, no? Arms around the neck type of thing. Such hugs I return by nuzzling my head against hers, maybe with a contented noise or an “'ello”. The bonus if that approach is that you don’t need to take your eyes off the screen 
If my girlfriend instead came around the side of the settee and glomped onto me, I’d hope she’d have the sense to look at the screen, as I only sit in front of one when I’m playing a game (we watch TV upstairs in bed), and she’s enough of a game player herself to know that you don’t mess with a person’s high score 
Either way, I wouldn’t ignore it - I’d either reciprocate or tell her to get off so I can pause and then reciprocate. Sometimes I fake apathy to a hug if I’m feeling put upon, but that’s because I’m an idiot and she knows it.