Hello Anne,
Welcome to Rootschat. Regarding Cambusnethan burials, here's what I managed to "dig-up" for you.
First of all, there were 2 burials in common Ground, that is - paupers burials. They are -
AGNES CAIN. Wishaw, aged 10 months.
Parents - William Cain and Mary Jane Cullen.
Interred in public ground 24th January 1872.
PATRICK CAIN. Wishaw, aged 23 months.
Parents as above.
Interred in public ground 19th September 1874.
You omitted to mention the wife of Peter Cullen who was born circa 1811, so I am guessing she could be Rose Flannigan. I am also going to guess that Peter Cullen born about 1840 may have married Mary Dale.
Did you know that the names Peter and Patrick are synonymous in Irish Gaelic? The name is Padraig. My guess is that both Peters were Padraigs and the Registrar used his own discretion in the translations. There is a good chance that neither of these men had any schooling, so they wouldn't be aware of the spelling.
Going on this assumption, I have included a Patrick.
I can now put the rest of the information into three, as I found three separate lairs or plots, each with Cains, Cullens or McCluskeys in them. They are all in the municipal cemetery at Cambusnethan, so there may well be internments after the dates listed here, the records I have finish at 1916 more or less.
Section C. lair 447.
WILLIAM CAIN. Coalminer, Wishaw, aged 61 years, married.
Parents - John Cain and Sarah McLaughlan.
Interred 1st September 1884.
ROSE ANN McCLUSKEY. Housewife, Wishaw, aged 30 years, married.
Parents - William Cain and Mary Jane Cullen.
Interred 9th December 1884.
ROSINA McALLISTER. Shotts, aged 2 years & 10 months.
Parents - Patrick McAllister and Mary Jane Cain.
Interred 7th January 1893.
MARY JANE CAIN. Wishaw, aged 74 years.
Parents - Peter Culman and Rose Lanyon.
Interred 29th November 1910.
Next lair is Section E. 128.
PATRICK CULLEN. Coalminer, Wishaw, aged 57 years, married.
Parents - Peter Cullen and Rose Flannigan.
Interred 3rd March 1899.
JOHN CULLEN. Coalminer, Wishaw, aged 19 years, single.
Parents - Patrick Cullen and Mary Dale.
Interred 26th January 1901.
JOHN DALE. Coalminer, Wishaw, aged 65 years, married.
Parents - James Dale and Margaret Golden.
Interred 8th March 1915.
Next and last lair is Section E. 1798.
ROSE ANN McCLUSKEY. Omoa poorhouse, aged 26 years, single.
Parents - John McCluskey and Rose Ann Cain.
Interred 2nd September 1908.
JOHN McCLUSKEY. Insurance Agent, Wishaw, aged 60 years, widower.
Parents - Edward McCluskey and Bridget McManus.
Interred 23rd May 1908.
Owner of this lair (1798 section E) was William McCluskey, garden labourer, 75 1/2 Young Street, Wishaw.
(The address was seventy-five and a half!)
I would read "Peter Culman and Rose Lanyon" as Peter Cullen and Rose Flannigan.
There may be a Poor Relief record for Rose Ann McCluskey who died in Omoa poorhouse. The records are now with North Lanarkshire Council at Lenziemill archives in Cumbernauld.
Omoa was the combination poorhouse for Cambusnethan, Shotts and Bothwell parishes, here is a good link with pictures.
http://www.workhouses.org.uk/index.html?Cambusnethan/Cambusnethan.shtmlHope this help you in some way.
Good luck with your research.
Lodger.