I am wondering what the reading and meaning of 公け is in the following sentence:
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It is pronounced おおやけ (although I've never seen it with け as okurigana -- usually just 公). It means "publicly/openly", similar to 公的 【こうてき】.
That sentence would translate as "Not wanting it to be made publicly known to everyone, He/I decided to break up with her in private."
The reading is