What with the internet and all, it's pretty easy to find basic information like this.
http://en.wikipedia.org/wiki/Earl_of_Errollwhich links to
http://en.wikipedia.org/wiki/Merlin_Hay,_24th_Earl_of_Errolli.e. the present Earl of Erroll was born in Edinburgh 1948 and is the chief of the Scottish clan Hay. His mother was Diana Hay, 23rd Countess of Erroll, and that seems to be where his Earl of Erroll title comes from, with a continuous location of Scotland.
You can click through the various past Earls of Erroll who have their own wiki pages, listed at the first link above, to see what might turn up.
Caroline Emma Hay's son John Short (Jr) married in 1768 so presumably Caroline was born sometime 1700-1725ish.
There really seems to be no connection to Yorkshire in the case of the titled Scottish Hays. Mary Hay, 14th Countess of Erroll
http://en.wikipedia.org/wiki/Mary_Hay,_14th_Countess_of_Errollgot the title when her brother died in 1717, and her sister's grandson got it when she died in 1758. She seems to be of the generation of either Caroline or Caroline's parents.
If Mary had had another brother, the title would have gone to him. If Mary's deceased brother had had children (he didn't), the title would have gone to one of them. And if she had sisters, the sisters' children would not have been named Hay, since there would have been no reason for a sister's husband to adopt the surname Hay. Ah, except ... read on, but it is still no help to you for Caroline.
Caroline could have been her younger sister, I suppose. But she wasn't:
http://www.rootschat.com/links/0svd/She had three brothers who died without children, and one sister, Margaret. Margaret had one daughter, Ann. It was Ann's son who got the title after Mary.
(Conflicting info about when Mary died, 1747 or 1758, but it doesn't matter for our purposes.)
If you read some of those links, you'll see that these Hays were Scots through and through, regularly getting imprisoned and beheaded by the English in the 1700s. There just isn't the least possibility that they were living in Yorkshire. Also, being Jacobites and so presumably fiercely Roman Catholic, it also is not likely that any of them would have been breeding CofE clergy by the mid-late 1700s.
The Hays that Caroline belonged to may well have been in Yorkshire, but would have been nothing to do with the Earl of Erroll people.
As for Ann Slate, it just makes no sense to light on two whose records are available somewhere and decide she must be one of them. There were undoubtedly Ann Slates all over England who simply have not shown up in any record you have access to.
All you can do in those cases, as is the case for many of us whose people didn't have exceptional names and didn't stay rooted in a little village somewhere for generations, is see whether Person X can be ruled out in some way (e.g. death in infancy, confirmed marriage).
But you are not likely ever going to be able to rule a particular Person X *in* since there just is no information on records in that era that would do that (e.g. father's name on marriage record, mother's surname or even given name on baptism record, age or spouse's name on burial record).
