The conjunction "a" is used to express contrast or to define a clause with a new meaning:
- У него синяя машина, а не чёрная: He has a blue car and not a black one
- Я обедаю, а не завтракаю: I eat lunch, not breakfast
- Сегодня вторник, а завтра будет среда: Today is Tuesday and tomorrow is Wednesday
- Река, а за ней ― город: A river and beyond it lies a city
I believe "but" should be considered correct here, so I have reported the answer "No, I am cooking soup, but not rice". "But" is used in English to express contrast, as in the example, "I went swimming today, but I didn't go running".
"But" can be used in all of mosfet07's English examples.
"He has a blue car, but not a black one"
"I eat lunch, but not breakfast"
"Today is Tuesday, but tomorrow is Wednesday"
"A river, but beyond it lies a city"
I think the best translation in this case is to not translate "а" at all: "No, I am cooking soup, not rice." That's a fitting answer to the question "Are you cooking rice?", which I imagine the Russian sentence is a response to. "No, I am cooking soup but not rice" rather seems to me to be something you would say as an answer to "Are you cooking (both) soup and rice?"
There was a lot of discussion about the sentence Мы не видим здесь места, wondering whether места was accusative plural or genitive singular. Duo translated it as singular, and the only way that could be valid is if не somehow negated место.
I realize that negation in Russian is extremely complicated, so much so that I don't think you can actually get it wrong if you pick accusative over genitive.
It seems like it's written as an answer to the question "Are you cooking rice?". It does seem like a strange sentence, and without the нет it would mean the same thing (just without the "No" in the beginning), but then it would also beg the question "why do I need to specify that I'm not cooking rice in the first place?" >_>.