Hello Liz.
I have had another search of the internet as there is more information being added all the time.

Just found on LDS.

Name Bernard Cain.
Event Type Birth.
Event Date 27 Jul 1893.
Event Place Brooklyn, Kings, New York, United States.
Gender Male.
Father's Name Bernard Cain.
Mother's Name Bridget Dougherty Cain.

And this surprise entry gleaned from the Internet which proves their Son Bernard was born in New York, United States.

Passenger: Bridget CAIN.
Date of departure: 24 August 1892.
Port of departure: Liverpool.
Passenger destination port: New York, USA.
Passenger destination: New York, USA.
Age: Adult.
Marital status: Married.
Occupation: Wife.
Passenger recorded on Page 3 of manifest.
The following children travelled with this passenger:
Fanny CAIN - 2 yrs (Frances, born 1889).
Infant CAIN - 2 mths (Margaret, born 1892).
Ship: TEUTONIC.

Averil