This scenario happens all of the time and one of the primary reasons is because people jump into answering the question without taking a second or two to figure out what they want to say. They feel the pressure to answer immediately so the reporter doesn't assume they are thinking of ways to evade the question.
Given the pressure-cooker environment of doing an interview, it's safe to assume that the better answer will almost always come to you when the heat is off and you have more time to collect your thoughts.
Breathe, Think, Speak
That said, you can still get a bit of an edge with this strategy I always recommend in my media training sessions: Pause before you answer. Just take a (silent) deep breath, collect your thoughts and ask yourself what you're trying to say in response to the question. And then answer. If it's a print interview, you've got an easy 4-5 seconds. For live TV interviews, you still have a couple of seconds before you need to open your mouth. Just remember that pausing only on the killer question doesn't work because it's too obvious.
Pausing is more effective than trying to buy time with statements like "could you repeat that again," "I'm not sure I understand your question" and, my all-time favorite, "that's a great question." You can get away with these once during an interview and beyond that they're a dead giveaway you're stalling.
The other thing that happens if you start answering before you know where you're going is you're just going to ramble until the light bulb goes off in your head and you figure out what the actual answer is. By that time the reporter may well have zoned out or interrupted you with their next question.
The ultimate danger with this type of talking and thinking outloud at the same time is you will inadvertently serve up some dishy information you'd never intended to divulge because your mind is overwhelmed.
I know those extra few seconds don't sound like much but they do help to get you on focus. And they will also prevent you from repeating negative ideas from the question that can become nightmare quotes.
So practice counting to three and then answering. It takes discipline but it's a behavior that you'll be glad you took the time to master.