Huge thanks for the help thus far. A small update.
I have contacted Carlisle records office who confirmed they have bastardy records for the Wigton area including Holme Abbey/Holme Cultram/Abbeytown (is it really the same place but has three different names?) but only for 1844 to 1856 so absolutely no use to me in this case.
I have purchased Hannah Brown's birth certificate and it confirmed what I thought: no father listed. I did glean her exact date and place of birth as 3rd April 1873 at Holme Abbey. Mum was Rachel Brown nee Brough a charwoman.
I am very keen to try to fathom what happened to my Hannah Brown. If indeed the Annie Brown aged 18 listed as a servant on the 1891 census in Plumpton, Hesket in the Forest, Cumberland is, in fact, my Hannah Brown as I suspect then I should surely either find a marriage or death before 1901 or an entry in the 1901 census. Does anyone have any ideas about how I can solve that one? If Hannah has any living descendants it may be useful to me to seek DNA tests but I can only do that if I can fathom what happened to her.
Finally, I noticed something I find rather odd. Rachel Brown nee Brough (the mother, my great great grandmother) died 19th November 1903. Probate/administration was granted for her at Carlisle 29 years later on 9th March1932!! to Rachel Brown (her daughter, my great grandmother). Can anyone suggest why such a delay? Apparently it was an administration not a will. I have often obtained wills and appreciate how useful they can be but is there any further information or document I can apply for other than being stuck with the extremely basic index information from the Probate Calendar on Ancestry showing the estate value of £145, 18s 2d?
As ever, any information, suggestions, discussion or advice is very warmly appreciated.