https://dictionary.cambridge.org/di...aditional/chase

to try very hard to persuade someone to have a relationship with you
追求，向…求愛
She's always chasing (after) men. 她總是追求男人。


https://www.oxfordlearnersdictionar...ter?q=run+after
run after somebody
​
(informal) to try to have a romantic or sexual relationship with somebody
synonym pursue

He's always running after younger women.