From the online family trees (you can check them too! -- free at mundia.com)
Minnie's father was George Conington 1856, a butcher
George's father was Edward Conington 1818, a butcher
Edward's father was Edward Conington 1766
Edward's father was Thomas Conington 1721
Only one brother is shown for Edward 1766: Francis 1760
Edward 1818 had brothers Thomas, William and Daniel
John Connington 1825 established the Conington Prize
John's father was Richard Connington 1796
Richard's father was James 1755
James's father was Jacob 1722
Only one brother is shown for James 1755: Jacob
(Jane, widow of James 1755, was still living in 1841 it seems, independent means;
Richard 1796 was a clerk, i.e. clergy, in 1841)
FS shows the parents of Jacob Conington 1722 as Jacob and Anne:
https://familysearch.org/pal:/MM9.1.1/JM8S-P6NThey also had childen Henry 1721 and An 1726.
FS doesn't show parents for Thomas Conington 1721 but it's always possible he was a brother or first cousin of Jacob Conington 1722.
So, assuming those are accurate, any connection there might be with Rev Richard Conington's family would be pretty remote (although they were undoubtedly related somehow).
If Thomas 1721 and Jacob 1822 were first cousins, Ethel/Dorothea? Conington and John Conington 1825 who established the prize would have been third cousins twice removed: grx4 grandchild and grx2 grandchild of the parents of Thomas and Jacob.
Just as my dad would have been the third cousin twice removed of the children of my second cousin four times removed, Viscount Sankey, if he'd had any children.

My dad had never heard of Viscount Sankey, and his children would never have heard of us -- but you never know, since in your case they were all named Conington and living in Lincolnshire, they may well have been well acquainted.
Since Dorothea was apparently in service and having a child before marriage at the age of around 28 ... well, she could have gone to uni first, but hmm, eh?