Hi
I have checked the church records for the Martin children for Our Lady and St Joseph Catholic Church which includes its predecessor churches. Below is a list of the children cross referenced their GRO registration.
Mary: Longtown Dec 1/4 1856 mmn Raggen.
born 25/8/1856 baptised 26/10/1856 mmn Hanley. God parents Simon Martin & Anna Mooney
Margaret: June 1/4 1858 mmn Hanley
born 13/5/1858 baptised 6/6/1858 mmn Regan. God parents Nicholas Martin & Julia Martin
Died November 1858
John: Dec 1/4 1859 mmn Hanley
born 07/11/1859 baptised 05/02/1860 mmn Regan. Godparents Thomas Martin & Julia Martin
Edward: March 1/4 1862 mmn Ragan
born 28/12/1861 baptised 02/01/862 mmn Regan. Godparents Julia Martin only.
Elizabeth: Sept 1/4 1866 mmn Hanley.
Born 02/07/1866 baptised 27/07/1866 mmn Hanlon. Godparents Jacob McCullock & Maria McCollings. (I'm sure that Hanlon is just a mistake and should be Hanley!)
Sarah Jane: June 1/4 1870 mmn Hanley.
Born 30/01/1870 baptised 19/07/1870 mmn Regan. Godparents Agnes Conway only
Hugh: Mar 1/4 1872 mmn Regan.
Born 31/01/1872 baptised 31/03/1872 mmn Regan. Godparents Patrick & Maria Metcalf.
As you will see the baptism for Ann born Sept 1/4 1864 is missing. I checked the baptisms prior to this date and well after this date but could find no baptism for a child to Patrick and Margaret of any name. The records are also indexed up to 1866 and I checked the indexes for Martin births and then checked the corresponding record to see if I had missed the baptism but with no joy. As all the other children are baptised within a short time from birth, I wonder if Margaret went home or to one of her relatives and the child was baptised somewhere other than Carlisle. I also checked the records for her baptism being listed under both Hanley and Regan/Ragan, but there were no baptisms under those names. Its just typical that the one you are most interested in, isnt there!!
As you can see there is no firm name for Margaret, as with the GRO births there is a mixture of Hanley and Regan, but I'm sure that she is the same person. There are not a lot of Martin baptisms, and judging by some of the godparents, they all belong to the same wider family, possibly brothers or cousins of Patrick?
I see that various trees on Ancestry have Julia Martin (wife of Simon) maiden name as McLauglin, but the baptism records show her consistently as Burke.
I dont know whether this has anything to do with your Margaret Martin, but there was a 'baptism' on 31/03/1861 for a Margaret (Margarita) Martin (DOB: 14/12/1841) recorded as sub conditiona, which i think means either a conversion to catholic or a baptism when it wasn't or arent sure it was done as a child.
Sorry the information that you wanted wasn't there but I hope the above may go on to help you solve the mystery. If there is anything else I can help with let me know.
regards
Julie