"The mayor eats the food with her hands."
Translation:Borgmästaren äter maten med händerna.
Sin/sitt/sina is used to refer back to the subject, while hans/hennes/dess is used to refer to yet another person. In the sentence above, you could use sina händer since it will then refer back to the mayor. But you can't say hennes händer, because that would mean she eats with somebody elses hands.
The notes for the "Medical" skill says: "Speaking about body parts, this is a good place to point out that in English, you always refer to your body parts with possessive pronouns. In Swedish however, we usually think that it is enough to use the determinate form of the noun. [...]"