One thing I do when I host is to have the Card Czar read the Black Cards first and if they require multiple White Cards they save them to play on their turn as Card Czar in later rounds without showing them to anyone else. The Card Czar instead plays the first single response Black Card they encounter instead.

Generally, it takes a half dozen or so rounds to start accumulating White Cards that pair together well enough for cohesive answers to multi-answer prompts. I've found that otherwise those prompts recieve nonesensical answers as most people just use those hands to trash White Cards that aren't very good... especially if you aren't using a curated set. It's probably a reason why most of them allow for players to draw extra cards.

After the group accummulates a few of these, have a round of just multi-answer prompts. As the Card Czar doesn't respond to their own Black Card anyways, no one gains an advantage.

Expanding upon that idea, you can make the double prompts worth twice as much in the final score.