I have quite a bit on the Cain family. Ephriam Frances Cain is my great-grandfather. His son, Richard Riley Cain, is my grandfather.
I have the same record on Lydia that you have presented.
Fact I just learned and will pass along in case you missed it:
The name Cain in immigration records appears 60% Irish, lesser numbers for Welsh and English origin.
I will gladly share any information I have that you may need.