Wouldn't it be "вони не п'ють кави з цукром."? Because кави is in the Gen. case because the sentence is negative?
Both should be accepted, I think.
There are no clear-cut rules, and the choice between genitive and accusative varies in different dialects.
For me, the option «ка́ву з цу́кром» makes ‘coffee with sugar’ sound like a single entity (вони́ не п'ють ка́ву з цу́кром = they don’t drink a drink called ‘coffee with sugar’), while genitive «ка́ви з цу́кром» somehow distances «ка́ви» from «з цу́кром» (вони́ не п'ють ка́ви з цу́кром = they don’t drink a drink called ‘coffee’ with sugar), although I can’t vouch the other speakers have the same distinction, or have any distinction at all.
Both "каву" and "кави" are correct in this sentence. What's unfortunate? Sorry I didn't quite understand your sentence :)