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.
To me, the pronunciation sounds exaggerated/artificial. If so, then we are being taught some separate language, made "easy" for foreigners. That would be unfortunate. I'd be interested to hear from more experienced listeners as to the extent to which the speech has been distorted.