Jenny comes(singing)a happy song.为什么用singing?

这个叫伴随状语,不是词组
comes和singing是同时发生的
所以也可以写成
Jenny comes and sings a happy song.
伴随状语表示与谓语的动作同时发生,通常可以改为用并列连词and连接的谓语或并列句.
I OFTEN SIT IN THE ARMCHAIR DEADING SOMETHING.
I OFTEN SIT IN THE ARMCHAIR and read SOMETHING.
具体看这里好了
当然,还有例外
He opened the door,sat in the sofa and turned on the TV
此时是三个动作,但是我们视作同时发生的,而且像汉语一样,最后个动作前加and