To become flustered or impatient and hurry; to be agitated or rushed (distinct from 咳込む, 'to cough').
締め切りが近づいて、彼はせき込んで仕事を進めていた。
With the deadline approaching, he was flustered and rushed through his work.
母にせかされて、朝ごはんをせき込んで食べた。
Pressed by my mother, I hurriedly ate breakfast.
バスに乗り遅れないようにせき込んで駅へ向かった。
I hurried off to the station so I wouldn't miss the bus.