Ballyconnell is in Tomregan(CI)/Killdalan(RC) parish.
I searched the Irish Family History Foundation's online database for Co. Cavan. Rcds B4 1840 are sparse. Many events weren't rcd'd, or rcds were lost.
For Bartholomew Gaffney, I found a B rcd in 1769 which may have been your Bart's father. Your Bart's birth wasn't rcd'd or has been lost. Bart's M was also not rcd'd, lost, or occurred in another county.
The earliest rcd'd Freehill B is in 1837 and M in 1855. Ann Freehill mayn't have been a local girl.
Assuming a marriage 1830ish, there are several Bernard Gaffneys whose births fit your time frame: 2 in 1834 (Kilbride parish), 1 in 1836 (Castlerahan), and 1 in 1841 Killeshandra). The main town for Kilbride is Mount Nugent.
For the other named children, I get matches on all of them in Kilbride parish. In fact, it looks like there are at least two families of Gaffneys in this parish. The only way to sort them would be by viewing the B rcds for each one and sort them by the mothers' names. It is also possible there was a duplicate transcription error.