Hi Anita,
I have searched the Civil Records for ALL of Ireland from 1864 to 1883, firstly to find a Marriage for Patrick Sullivan & Margaret Donoghue and did not succeed Secondly I again searched All of Ireland for a Marriage of a Margaret Donoghue & any Sullivan for the same range of years and only came up with one record of Jeremiah Sullivan & Margaret Donoghue in Enniskeane Parish in 1870 Link to their Civil Marriage below,
Below this Civil Marriage is a Baptismal Record from Enniskeane Church for Margaret.
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1870/11378/8177328.pdf
https://churchrecords.irishgenealogy.ie/churchrecords/display-pdf.jsp?pdfName=cork%20%26%20ross.enniskeane.p4798.00269
If Annie was 20 in 1903 then she would have been born/baptised in 1881 - 1883, there is no record of an Annie Sullivan born in Ireland with Sullivan/Donoghue between these years.
I have found a church record of a Baptism a Margaret Sullivan with Parents Jerry Sullivan & Margaret Doinoghue in Dunmanway Church dated 16 March 1876 and while the first name is Margaret, there is an Annie Sullivan as a sponsor and it is not unusual for errors of this nature in Church Records. There is an Irish saying that the person who made the entry error had been standing too cliose to the altar wine. The church record is here:
https://churchrecords.irishgenealogy.ie/churchrecords/display-pdf.jsp?pdfName=cork%20%26%20ross.dunmanway.p4805.00738
The transcribed version of it is here:
https://churchrecords.irishgenealogy.ie/churchrecords/details/b3b8cd0175071?b=https%3A%2F%2Fchurchrecords.irishgenealogy.ie%2Fchurchrecords%2Fsearch.jsp%3Fname2fm%3DMargaret%26name2l%3DDonoghue%26namefm%3DAnne%26namel%3DSullivan%26location%3D%26yyfrom%3D1870%26yyto%3D1885%26submit%3DSearch%26sort%3D%26pageSize%3D100%26diocese%3D%26parish%3D%26century%3D%26decade%3D%26ddBfrom%3D%26ddMfrom%3D%26ddDfrom%3D%26mmBfrom%3D%26mmMfrom%3D%26mmDfrom%3D%26yyBfrom%3D%26yyMfrom%3D%26yyDfrom%3D%26ddBto%3D%26ddMto%3D%26ddDto%3D%26mmBto%3D%26mmMto%3D%26mmDto%3D%26yyBto%3D%26yyMto%3D%26yyDto%3D%26locationB%3D%26locationM%3D%26locationD%3D%26member0%3D%26member1%3D%26member2%3D%26member3%3D%26member4%3D%26member5%3D%26member6%3D%26member7%3D%26member8%3D%26member9%3D%26namef0%3D%26namef1%3D%26namef2%3D%26namef3%3D%26namef4%3D%26namef5%3D%26namef6%3D%26namef7%3D%26namef8%3D%26namef9%3D%26namel0%3D%26namel1%3D%26namel2%3D%26namel3%3D%26namel4%3D%26namel5%3D%26namel6%3D%26namel7%3D%26namel8%3D%26namel9%3D%26keyword%3D%26event%3DB Sorry for large link.
Again we have a mystery as the only Margaret Sullivan with a mother formerly Donoghue to be found in the Civil Register for 1876 is dated in June of that year but one could note that Jeremiah as the informer is illiterate as there is only his "mark" as the informant and informer and this may also have been the reason that Margaret was in the Church Record and not Annie. There are hundreds of errors such as these on bot Civil and Church records.
The Civil record of the birth is here:
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1876/03049/2118127.pdf
Anita, I wish I could be more positive but in all I have spent about eleven hours on this query and am absolutely certain that there are no other records coming even remotely nearer to the facts as you know them. I know that the age of Annie is greater than that she stated at her marriage but she is the only daughter of a Sullivan/Donoghue couple in the whole country and after years of experience in tracing people, I would be very surprised if this is not her. Just one other little thing, you were very confident that Macroom was the birth place of Annie, Dunmmanway which was given as the birthplace of Jeremiah who could have been Jeremiah Patrick and Dunmanway is just about 17 miles from there and he and Margaret were married in Murragh Civil Parish in Enniskeane which indicates that Margaret was born in that area which is even closer to Macroom at 15 miles. Again possible that both the name and area were mistakenly written down or misheard between father, mother and daughter.
My apologies for all the rambling explanations but I feel very confidant that these are the correct people and want you to keep the points above in mind in any future investigations. Please also remember that the information given is based on the records of Ireland as a whole rather than just Cork County. You can find all of the above information on www,irishgenealogy.ie
All the very best,
Hope I helped you find your past.
Frank