其实并没有那么复杂,你可以将它变成一般句,例如你的例子,do you think 在这里可以看成一个插入语,它可以放在任何位置,也可以省去：
Who the public might choose ,do you think,as their favorite singer this year?
Who the public might choose as their favorite singer,do you think , this year?
Who the public might choose as their favorite singer this year,do you think?