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I'm concerned about my wife's dry cough.
Literal
Wife [genitive-の] dry-cough [subject-が] is-concerning.
A short worry. 乾咳 (からせき, sometimes かんがい) is 'dry cough' — 乾 ('dry') + 咳 ('cough'), a specific medical descriptor distinguishing dry coughs from productive/wet coughs (湿性咳 or 湿った咳). 気になる is a key set expression literally 'becomes a concern,' meaning 'is concerning,' 'is on my mind,' 'bothers me.' It covers both negative worries and neutral attention-catching things. Note the が particle marking the subject of 気になる — not を, because 気になる treats its object-like argument as a subject (literally 'X becomes a concern [to me]').