Statutory birth registration started in Ireland on 1.1.1864 so you should be able to find birth certificates for all the births after that date on this site:
Note that the father’s name was Hugh (not High). I found the following births:
Bridget Harren 9.4.1864 at Killycreen
Ellen Haran 25.5.1866 Killycreen
Rose Haran 21.10.1870 at Killycreen
Charles Joseph Harren 28th May 1878 at Killycreen, father was a farmer.
I did not see a birth for Teresa in 1878. Charles wasn’t a twin. He was born in May so it wouldn’t really be possible for him to have had a sibling in that same year. All the births were at the farm in Killycreen. (Holywell which is evidently in some records is the name of the local registration district).
Hugh died some time between 1878 and 1901. I can’t see his death in the records.
This looks to be Bridget and son Charles Joseph in the 1901 census:
There was another Harren family next door who are presumably related:
1911 census, Charles has married Alice Dolan (on 21.10.1907) and his mother’s age has become a bit more realistic (often due to the arrival of the old age pension in 1909 for over 70s):
There was a death on 11.4.1905 at Killycreen of Hugh Harren, aged 38, occupation draper. He was a bachelor. The informant was his brother John.
Griffiths Valuation for Killycreen East (1862) shows Hugh & Patrick Harney (sic) jointly on plot 5 which was a 91 acre farm.
Probate abstract from PRONI wills site:
Probate of the Will of Patrick Harran late of Killycreen County Fermanagh Farmer who died 28 December 1907 granted at Armagh to John Harran Farmer.
The above will is on the PRONI wills site and can be read there free. It mentions his wife Lucy and son John.
Hope this helps a bit.