Hi Laurie,
Chances are the children would have been sent to the Bridge of Weir, Quarrier Home in Renfrew, Scotland and then on to the Fairknowe Home in Brockville, Ontario. My great grandfather and two of his siblings did the same voyage but in 1892. I have tried to get information about them once they were in the home in Canada, but apparently the records were destroyed in the 1930's.
What year was George and sister born in? Are you sure they were born on the island as there is a Jessie and George Ashworth listed in the British Census of 1881 in York England with mother MAry. No father listed, could this be them?
Sandy
Mary ASHWORTH Head W Female 29 Kirkowen, Scotland
Jessie ASHWORTH Daur Female 3 Hull, York, England
Reuben ASHWORTH Son Male 1 Oldham, Lancashire, England
K. ASHWORTH Lodger M Male 32 Oldham, Lancashire, England Cotton Factory Hand
Eliza ASHWORTH Lodger M Female 31 Scotland
George ASHWORTH Son Male 1 Hull, York, England
Violet ASHWORTH Daur Female 1 m Hull, York, England
Tom MOSS Lodger M Male 49 Oldham, Lancashire, England Machinist 72

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Source Information:
Dwelling 126 Cumberland St
Census Place Sculcoates, York, England
Family History Library Film 1342149
Public Records Office Reference RG11
Piece / Folio 4758 / 80
Page Number 12