Question

the student always have fun singing English songs. 为什么是singing

Answer 1

singing English songs你可以当作状语，主语重复时可以动词ing形式充当状语，还原状语从句就是The student always have fun when he(she)sings English songs.

另外，如楼上所述， have fun doing sth.意思做某事很开心。

Answer 2

HAVE FUN DOING STH 是固定的句子

Answer 3

have fun doing sth. 开心地做某事
没有have fun to do sth结构