What does "air" mean in the following context?
She was referring to Mr Church and the air of mystery that clung to him.
"Air of mystery" is an idiom meaning aura or atmosphere of mystery. That is, Mr. Church had an aura of mystery about him.
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"Air of mystery" is an idiom meaning aura or atmosphere of mystery. That is, Mr. Church had an aura of mystery about him.
Sense 2 at https://en.oxforddictionaries.com/definition/us/air :
An impression of a quality or manner given by someone or something.