。
Yoko avoided answering my question.
Literal
Yoko [topic-は] my [possessive-の] question [to-に] answer [thing-こと] [object-を] avoided.
答える ('answer') takes に for the question being answered (parallel to 会う takes に for the person met). こと nominalizes 答える into 'the act of answering.' 避ける ('to avoid') takes the nominalized clause as its object.