expression, noun
awkward question; difficult question
A question that is hard to answer, often because it is embarrassing, sensitive, or requires a delicate response.
彼女に「どうして別れたの？」と聞かれて、答えにくい質問だった。
When she asked me, 'Why did you break up?', it was an awkward question.
面接で答えにくい質問をされて困った。
I was troubled when I was asked a difficult question in the interview.
難しい質問 is a general 'difficult question' (intellectually challenging), while 答えにくい質問 emphasizes the awkwardness or emotional difficulty of answering.
Compound of 答えにくい (hard to answer) + 質問 (question). The phrase is transparent and follows standard Japanese grammar.