To worry or fret; to be anxious and uneasy about an outcome (an idiomatic expression).
試験の合格発表を待っている間、家族みんなが気をもんだ。
While waiting for the exam results, the whole family was anxiously fretting.
患者の容体が良くなるかどうか、医者も家族も気をもんでいる。
Both the doctor and the family are anxious, worrying about whether the patient's condition will improve.
結果が出るまでは気をもむなと言われたが、つい心配してしまう。
I was told not to worry until the results come out, but I can't help being anxious.