Looking for my great grandmother's birth record - Catherine Dunn(e)

Hi! I have been searching for my great grandmothers birth record as well as information on her father for some time now so any help that could be provide would be a greatly appreciated. Here is some information about her:

Name: Catherine Dunn or Dunne (I have seen it spelled both ways)

Born: November 6th 1899 or 1900 in Cork might be linked to Aghada

Mother: Ellen Dunne was born May 1st 1872 in Cork Gyleen

Mother married a man named William Sick at some point

I know my great grandmother Catherine was living with her grandparents Samuel and Johannah Dunn(e) in Glanturkin (Gyleen Town) (Inch, Cork) in 1911 as I have found them on the census. The census lists as Catherine as Samuel and Johannahs niece which I know to be untrue as Samuel and Johannah are Ellens parents making Catherine their granddaughter.

My great grandmother arrived in America on June 19th, 1920 at the age of 19 on the ship manifest it states that her closest relative in Ireland is her mother Mrs. Ellen Sisk Guileen, Whitegate Co Cork.

I have neve been about to locate any birth or baptismal record for my great grandmother and I do not know who her father is. It is possible that William Sick is in fact her father, but I do not really believe that he is.

The thing that is wired to me is that I was also able to find her mother Ellen and William on the 1911 census in Glyleen Town (Corkbeg, Cork) and I wonder why she was not living with her mother.

My grandmother never talked much about her time in Ireland. I never heard he speak about her mother. The only thing she ever told me was she could remember going down to see the Titanic when it docked off the coast. Later on, right before she passed away she mentioned Samuel (who I now know to be her grandfather) however the only thing she said was how much she hated him.

Thanks for your time.