Hello again,
it is all very difficult because of the O'Hagan/Hagan plus spellings and transcriptions and the missing people in 1901. However, I will try to outline what I think is the case.
I also think that 1911 census is perhaps crucial to your enquiries.
Do you have a published tree or access to those trees with the family information?
FREE BMD
1883 Marriage- Terence Edward O'Hagan and Annie McQuade
1891 Edward, Annie and Annie S O'Hagan -Birkenhead
1901 Terence E O'Hagan, Anne and several children in West Derby
I think it is this Annie who dies in 1910.
1911 free indexes shows Terence Ohagan in Birkenhead
You need to access the 1911 census if you wish to find out about Terence and his family. I do not think they are your family.
1901 is a problem- I can't see anyone of your Edward O'Hagan family
Free BMD
Marriage 1903 Edward Francis Hagan and Annie Smythe Liverpool
Birth 1901 Josephine O'Hagan Birkenhead
1911 free index has a Josephine Hagan 9 yrs in Birkenhead
Using the 'other persons in household' search there is an Edward F, 41 yrs
Again you need to look for this family in 1911 to see other children and places of birth to check your family.
The ages of Terence and Edward are not the same and looking at their families, there are different children.
Do you have any of the marriage certificates of Edward Francis?
I hope you can follow this. The names are important and 1911 will be vital to you.
regards
heywood