Most people received public burials at one time it's only really in the 20th century that private burials became more common. It wasn't necessarily because of limited finances. A decent Christian burial was all that was expected. Someone would have paid for just the burial at Kirkdale. Pauper burials were different they took place at the parish cemetery and were paid for by the parish. At that time paupers were buried at Walton Park. Robert Tressell is one famous pauper burial at Walton Park. In his case his family had money but they asked the parish to bury him so even with pauper burials it may not necessarily reflect the person's situation.
Blue
Sorry but you are confused about private burials churchyards contain far more private grave plots than public grave plots most graves in churchyards are in fact, if not in name private plots.
In the early 19th century many cemeteries were opened to provide an alternative to the overcrowded church graveyards. I suggest your confusion may stem from the fact that when cemeteries opened the more wealthy purchased private plots in the cemetery but the poorer classes tended to still be buried in the churchyard, skewing the figures. Having said that most cemeteries of the 19th century had a three class system of graves (First class, Second Class, Public).
The public graves being interspersed between private graves to give a less crowded appearance to the cemetery.
It is interesting to note how public graved changed in the 19th century, for instance in Leeds General Cemetery, Woodhouse Moor (opened 1835) the first public grave in the consecrated section was dug 14ft deep and contained 8 bodies, by 1837 the average was 11 bodies but in 1838 the occupancy was changed to 18 and by June 1839 24 bodies per public grave, however in the unconsecrated section the average was 5 bodies in a public grave.
I have not done any research to see if this could explain why some members of an otherwise church going families were buried in unconsecrated sections of cemeteries.
Cheers
Guy