So ''Хочешь супа?'' specifically means ''Do you want some soup?'' whereas ''Хочешь суп'' would mean ''Do you want the soup?'', is that right?
This is about how it works. English also has "Do you want soup?", which works for both.
No one forces you to use these structures, of course. However, in Russian asking questions like «Хотите чая?» or «Ещё сока?» is quite common.
It is also very common to use "будешь?" instead of "хочешь?"
- Ты будешь суп? - Да, с удовольствием! / literally: Will you eat soup? - Yes, with pleasure!
- Ты будешь кашу? - Нет, спасибо. / literally: Will you eat porridge? - No, thanks.
Are there such examples in the course?
Yes, супа is in genitive case. Genitive case is used often for quantities and to express "some". Ideas like "some tea", "a little wine" ect, take genitive case. You would use accusative case for sentences like Мы едим суп. (We eat soup), when soup is the direct object being eaten.
Would, "Want soup?" be incorrect? Or is this more formal so you would say "do you want soup?"
People say this translates to being "some soup" due to the genitive ending, but it seems to me that the ending is a consequence of it being a direct object in accusative.. So isnt this simply; "Do you want soup?" instead of "Do you want some soup?"