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My wife was adopted. A number of years ago she discovered who her mother was due to the adoption society making contact with the mother's family. But too late as her mother had died six months earlier. My wife subsequently got to know her half-siblings and her aunts and uncles but all claimed that they had no knowledge of who her father was. The aunts and uncles (ie her mother's siblings) have now all died and if they did know who the father was have taken this to their grave.
My wife has since taken an Ancestry DNA test and persuaded one of her half-sisters to do likewise and she has now eliminated all those people who they have in common so she is now left with a list that presumably are on her father's side. The problem is that the nearest match is fourth cousins. She has put the results on Gedmatch but to be honest neither of us really understand DNa. Has anyone any suggestions where to go from here?

My wife doesn't know who her father is but would desperately love to find out. She has done an Ancestry DNA test but this has thrown up nothing closer than 3rd cousins.
Can anyone advise how she should proceed.

First of all, apologies if this is the wrong place to post this question but I thought it the most suitable.
I have traced my 2x Gt grandfather to Bermondsey in Surrey where he got married in Oct. 1855. The marriage certificate shows his father James was from Co. Kilkenny in Ireland. It would appear that the Kennedys migrated to London during the Irish potato famine. Unfortunately the marriage cert. does not identify what part of Kilkenny he was from. Being Irish Catholics it would be very unlikely that John was an only child but unfortunately I don't have the names of any siblings. Can anyone suggest how I might go about trying to trace siblings? They could have been born in either Kilkenny or Bermondsey or indeed both places.

I have traced my gg grandparents to Bermondsey where they got married 7th October 1955. My gg grandfather was John Kennedy born circa 1830. My gg grandmother was Ellen Crowley born circa 1835. According to the 1861 census John was born in Co. Kilkenny and Ellen in Co. Cork. John''s parent's  names were James and Elizabeth (maiden name unknown ) and Ellen's parents Jeremiah and Mary (again  maiden name unknown ). I am unaware of any siblings but being Irish Catholics the probability  is high that there were some. I am unaware where in Co. Kilkenny John Kennedy came from. Kennedy is a very common surname in that part of Ireland. Also Ellen Crowley. Again I don't know what part of Co. Cork Jeremiah Crowley came from. Crowley is also a very common name in Co.Cork as it is here the name originates.
I have now reached a block in my research and would welcome any help in progressing my research.

Sorry if I have posted in the wrong section but thought this the most appropriate. My father worked with the admiralty I think during WW2. Is there somewhere I can check to see if this is the case? His name was James (Jim) Kennedy dob 11 Apr 1918?