Translation:The cook cooks with any ingredient.
The problem is that "whatever" can be a pronoun or an adverb, not an adjective. So it's the English that would be wrong if you said he cooks with "whatever ingredients." If you could modify it to "whatever ingredients he has at hand," then it could fly. But those extra words aren't necessary in the Italian variation. Ah, the richness of languages!
I think that "any ingredient" and "any ingredients" have slightly different senses (in English). "any ingredient" means for each (single) ingredient he can find a way to cook with it. "any ingredients" means that for each set of ingredients he can cook them. I'm not sure if the senses are the same in Italian, but unless there's a specific rule (alcun and nessun always take singular for instance), I'd match the number (pluralness) of the Italian word (here ingrediente not ingredienti) to the English word.
If I interpret Collins correctly whatever ingredient should be equally correct as any ingredient: https://www.collinsdictionary.com/dictionary/italian-english/qualunque
mzmalvina- I think it could, especially if a phrase like "...with whichever ingredient 'is available'" is added. As it stands I think there's a difference between the two: 'any' is broader and would presumably include the option to use any one of all ingredients while 'whichever' is somewhat restrictive, meaning there's a narrower choice available.
Acutally, "any" is used when there is doubt about the existence of the commodity. That is why saying that it is for negatives and questions is usually a good explanation. But if, for example, you offer someone some coffee, it will be a question with "some" (Do you want some coffee). That is because in this case I know I have the coffee! If I cannot find the coffee, I might ask my husband, "Do we have any coffee?" as its existence is negligible. (I hope that makes sense, and btw I think I need a coffee now!)
in this same exercise, another prompt was "qualunque verdura" and it translated as "whichever vegetable", yet "qualunque ingrediente" cannot translate to "whichever ingredient". Not sure whether this is just an oversight on DL''s part, or there is some subtle difference which I'm not aware of . Reported.
I think alcuno/alcuna/alcun (singular) is almost always only used in negative sentences. e.g "Non ho alcuno dolore" = "I don't have any pain", "Non ho alcun desiderio di discutere con voi."= "I have no desire to argue with you", " Treccani, however suggests that it can sometimes be used in positive sentences: https://www.treccani.it/vocabolario/alcuno. In any case, "ingrediente" is plural so you would have to use "alcuni" (plural) which would be fine in a positive sentence.