Question

What does elizabeth ask hooper to do in order for her to stay engaged to him?

Answer 1

I believe she asked hooper to uncover his black veil.
But hooper still couldn't managed to fulfill her request, and the engagement between the two of them finally canceled.
Elizabeth couldn't get around her head what kind of insercurity that hooper got to the point were he couldn't trust her to uncover the veil.

Answer 2

In order for Elizabeth to remain engaged to Hooper, Elizabeth requests that he remove the veil. He goes on to tell her that if he removes the veil, it will remove all the darkness that lies between their souls.