Irish Grandma

Hello 

I am searching for info on my grandma and great grandparents. We believe grandma was born in May 1913, called Frances Armstrong, possibly born to Mary Armstrong and James Armstrong. I have found multiple workhouse admissions for Mary, Frances and George. Frances had an older brother called George who we believe was born in 1908. George’s daughter provided us with the birth certificate he used - however we do not know if it is the correct one / how he came to think / know it was his. It states he was born January 11th 1908 to James and Mary (nee Banks) Armstrong in Shears Street in Cork - I have been unable to find a marriage record for Mary and James in Ireland, Scotland or England. Two workhouse records state that Mary’s husband was James and he was in Baggot Street Hospital, another record states that James is in prison. Frances never had a birth certificate, unsure if her birth was never registered or if she was registered under a different name. The majority of their workhouse admissions are in Dublin, some are in rathdown, the first workhouse record I have found showing Frances is in July 1913 and states she is 7 weeks old. There is an admission from May 3rd to 8th 1913 for Mary and George which states that Mary is pregnant. George was taken into care around the age of 6 and Frances into care around the age of 3, where they stayed until they were young adults. Frances married my English grandad James Bate in Belfast on September 26th 1939, sometime after they moved to Leigh, Greater Manchester England where grandma had three daughters - Doreen born in 1941, Gillian born in 1945 and Barbara (my mum) born in 1949, Frances died on October 21st 1985 in Leigh. 
Both mum and I have done ancestry DNA tests but our closest maternal match is only 130cms, we have the makings of a family tree albeit we know nothing about Frances’ family. I have uploaded our results to Gedmatch, My Heritage, FTDNA. I have contacted the GRO, National Archives Ireland, Smyly Trust, no information found by any of them. Grandma was Protestant - not sure if she would have had a baptismal record but havent been sure where to search either - the majority of the workhouse records show them as homeless - so not sure if she would have been baptised officially. I have looked at 1901 and 1911 census, one record for 1911 shows a Mary and George Armstrong in the Tiperrary workhouse, there is also a birth record for a George Armstrong on January 7th 1908 in Cork workhouse to a single mum Mary Armstrong. Ancestry has automatically labeled Frances and George as half siblings, the BC we have for George could be a red herring and they could both have been born to Mary Armstrong with different fathers. 
Thankyou for taking the time to read this 

Rebecca