to chase; to run after; to pursue — to follow rapidly behind someone/something, or to pursue goals/dreams.
犬がボールを追いかけて走った。
The dog ran chasing after the ball.
彼女は夢を追いかけるために都会へ出た。
She went to the city to pursue her dreams.
子どもたちは公園で風船を追いかけて遊んでいた。
The children were playing by chasing balloons in the park.
shortly after; following that — used in expressions like '追いかけて' or '追いかけるように' to indicate something happens right after.
スピーチの後、追いかけるように質問が続いた。
Shortly after the speech, questions followed one after another.
演奏が終わると、追いかけるように拍手が起きた。
When the performance ended, applause rose immediately afterward.