I do find it weird that a HEFFERNAN appears on my great grandfathers birth record and later his mother leaves his father Thomas PLANE and married a HEFFERNAN.
Perhaps you have information to the contrary, but I do not see a marriage between PLANE/GALLAGHER and HEFFERNAN. Where and when do you believe it took place?
Maybe it was after the death of Mary's husband, Thomas, in 1937, so not accessible in the NZ BMD.
I can see from records the presence in Australia of the sons Frederick James and Charles Eric PLANE. Perhaps she did not take them when she left with HEFFERNAN.
Do you know what happened to Henrietta?
It will be interesting to hear whether the information we have found is of interest to you
Sue