Affection WITH something
Now, I know that having affection FOR something is correct, but I'm not sure whether or not the word FOR could be replaced with the word WITH, without making the sentence sound odd to a native ear.

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no, definitely not 'with'.  When it comes to liking and dislike, You have 
an affection for,
 an affinity with,
an aversion to
a liking for
a fondness for
an attraction to
that's about all I can think of for now.  Hope that helps

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You could you could show affection for someone with a hug.

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No, that wouldn't be right. You can be afflicted with something, but of course that has totally different meaning.