Perhaps you have already found the information you are looking for, Sara, but here is what I have on Arthur's children. I found them in the 1881 and 1891 British census. I am actually currently looking for the civil records of the births of the last two. The first two are in the Kirk Michael Parish Christening registers, which I ordered and viewed just last week.

Arthur married Mary Ann Shimmin of Kirk German
They had 4 children, all born in Kirk Michael:

Eliza Ann 26 July 1874

Cecilia Maragaret 9 Dec 1876

Ethelreda 1878/79

Alfred Arthur 1881

If you would like more info of this part of the Moore family, I'd be happy to share as much as I know. Good luck.