To Chase
To Pursue, To Run After, To Follow
transitive verb, ichidan verb
追う means "to follow" or “to chase” and かける in this case means “to run,” so this word means to chase, to pursue, or to run after someone or something.
The reading comes from 追う as well as the kanji reading you learned too.
マリのこと、追いかけるつもりはないよ。
I have no intention of pursuing Mari.
あいつ、パスポート忘れてる！今すぐ追いかけなきゃ。
He forgot his passport! I have to go after him right now.
あそこで、ウサギを追いかけるライオンを見ました。
I saw a lion chasing a rabbit over there.