I grabbed a swab to try helping her out. In the end, Boadi managed to clear the carrot by herself. I think picking her up to get a better look at her teeth must have worked it loose.
It's rare for a guinea pig to get food stuck like this. Eclair once jammed a sunflower seed in her teeth (before I knew that I shouldn't feed seeds to guinea pigs). I also saw this kind of behavior when Bertie developed malocclusion; food can get trapped when the molars are overgrown. I knew Bertie needed to go to the vet when she started making faces like this and dropping bits of food when she ate. It may look silly and cute, but if you see your guinea pig making faces like this regularly, it's time to have their teeth checked out by the vet. Boadicea's incisors are nice and straight, and I haven't seen her do this again. I bet she inhaled too large of a chunk too quickly and it got caught in her cheek. Cheeky pig!