Have you seen GenesReunited which has an entry 
Mary   Gunning   1876   Newtownards, County ...   treeholder Demi {Anderson} 
to contact you would have to join to see his or her tree    
Thoughts have you joined 
https://en-gb.facebook.com/Newtownardsfamilyhistorygroup/ its free
usaPetticrew wondered about session books at Presbtyerian Church, Greenwell Street, Newtownards.    I sadly think not,  and also the marriage notice books started up in 1879  just a couple of years too late perhaps see 
https://www.nidirect.gov.uk/sites/default/files/publications/Guide_to_church_records.pdf for Presbtyerian Church, Greenwell Street, Newtownards
Baptisms, 1866-99 and 1946; marriages, 1870-1903; marriage notice books, 1879-1906; index to marriages, 1870-1936; list of members, c.1920; communion roll, 1909-10. 
I found an Esther Gunning marrying a William XXXX in Newtownards on emeraldancestors but found it dated 1860. it made me wonder had Anna McCafferty b 1857 later had met perhaps Thomas Gunning family perhaps including a sister ?
1901 Census Ireland 
http://www.census.nationalarchives.ie house 104 in Greenwell Street (Newtownards Urban, Down)
Gunning   Mary   25   Female   Daughter   Presbyterian   County Down   Power Loom Weaver   Read and write   English   Not Married   -
Gunning   Annie   46   Female   Head of Family   Presbyterian   County Down   Embroider   Neither read nor write   English   Married   -
 I thought I would try a draft tree as becoming confused 
McCafferty children 2 brothers
1.  Henry McCafferty 
children
1.1.  Jane McCafferty 1814,
1.2.  Henry McCafferty 1817
1.3.  Ann McCafferty 1819, 
1.4.  Henry McCafferty 1821 or his father above living in Mill St  [Griffiths valuation1863]
1.5.  Martha McCafferty 1825      
2.  Robert McCafferty m Jane Osborne
children
2.1.  David Osborne McCafferty 17 March 1822 bapt’ ‘Kilinchy Church,
2.2.  Anna McCafferty 20 June 1824 bapt’ ‘Kilinchy Church
2.3.  Jane McCafferty 12 March 1827 bapt’ ‘Kilinchy Church
2.4.  Robert McCafferty 10 May 1829(Anna's father) bapt’ ‘Kilinchy Church m 29 November 1845 Newtowards Esther/Easter Hollinger
.....children possible more syblings
.....2.4.1.  Anna McCafferty b1857  Annie in 1891 census
.........children.
.........2.4.1.1.  William James McCafferty 18th March1873 bapt’ Presbyterian Church 
Greenwell St Newtownards father unknown if known as William J Gunning in 1881 and 1891 see England, Wales & Scotland Census Bridgeton, Glasgow
.........2.4.1.2.  Easter/ Esther McCafferty {?}/could be Gunning? 1st Nov 1874
.........2.4.3.3.  Mary McCafferty {?} between 1875-1877 1876      census 1891 as Mary Gunning is listed on GenesReunited by tree holder  Demi Anderson have I got the right family?
.....2.4.1.  Anna McCafferty/ Anna McAffirty/Annie? m 23 Jul 1877 Greenwell Street Presbyterian Church  Newtownards ,Thomas Gunning b 1855 shoemaker...children?
in 1881 living in Preston St Bridgeton, Glasgow...have I got the right family?
in 1891 living in Main St Bridgeton, Glasgow  ...have I got the right family?
2.5.  John McCafferty 1839 unsure
2.6.  Agnes McCafferty 1841 unsure 
Looking at this draft uncorrected tree I wondered if Easter/Esther had  died young eg aged 1-3 years old. That event might well have brought parents closer together and a reason they married a bit late. Finding apparently no burial records have been kept at Greenwell St
Presbyterians were sometimes  buried in other P church graveyards   eg burials, 1892-1901. at Regent St P church, even sometimes were buried in Church of Ireland graveyards not knowing where to start I tried to check Old and new  Movilla but got lost in finding the thorough way. That must be done for Easter
 the other two children William James and Mary appear to have taken the name Gunning then or soon after.  Have you looked for Mary baptism at greenwell church b 1876? say a couple of years later.    I only realised late that Annie and daughter Mary Gunning were living in Greenwell street itself in irish census. Apart from PRONI holding church records, familyhistory.org did an amazing qty of microwork , they would be free.    
If you joined The North of Ireland Family History Society (NIFHS) their website says There is a free Look Up Service available for Members who are unable to visit the Library.  Looking for furtherafield sources JohnGrenham.com  ‘s  website gave me the idea could the McCafferty in Belfast be the relatives who adopted her.  
JohnGrenham.com  ‘s  website gave me the idea could the McCafferty in Belfast be  the relatives who adopted her.  
Have you been in direct contact with Greenwell st church. If you haven’t, My uncle found enclosing a £5 note for church funds with his opening request could they help letter, oiled the works best, when absolutely stuck. He also got requestd for more money on one occasion for the work the clergyman would undertake only if paid. So over to you.
Other than that I run out of suggestions except producing a one person search listing all the sources physically carefully looked up with each named variant  of spelling with the overall purpose of asking readers have you missed any source.  We are back to the sources notonline among which include many church records for different reasons.
much good luck needed onwards