I looked on Ellisisland.org for Ellen Moore but couldn't see one arriving in 1910. I found an Eleanor Moore arriving in 1910, but note that you need to register to view the original. I think it is free though? Am a bit pushed for time atm so have no time to go through all the processes and not much point joining just to view this one record. I am wondering if you would be able to post on here a snip of the place of birth, with enough other writing so we can get a feel for the style and perhaps interpret the place name.
Although you say it is illegible, someone else
may be able to decipher it for you. It's worth a try anyway. Is there a help desk, or 'contact us' whereby you could request a more clear scan of this particular record?
If you are able to provide a snip, it would be worth starting a new thread on the 'Deciphering" board to specifically request suggestions about place name. Provide a link to this thread to give helpers some background to your request.
I have no experience with research in the USA. Have you posted a request for help finding her marriage on the USA board? (once again supply links to related threads)
I'm sure she is one of the dozen or so Ellen Moores on the 1891 and 1901 English cenuses - it's just a matter of finding a clue as to her father/parents names.
One more thing, do you have Ellen's death certificate? I am wondering if American death certificates would name her father/parents? Or perhaps an obituary may give some information about her life? Less likely perhaps, but might she have left a will? It may name siblings/other family members who remained in England?