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This is a sentence from 「ふたり」 by 赤川次郎
Does it read as むこう？If that is the case, then what is the difference between 向う and 向こう？
This is read as むこう.
向こう is the "standard" spelling, but people (especially novelists like him) sometimes use nonstandard or customary spellings like this.
You can see the list of such kanji here:
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