Charles Darwin wrote a letter to a man pleading with him to read his book on evolution, but the man refused to do. A copy of the letter is on display at (I think) the Yorkshire museum. I have the names of John Phillips and Strata Smith. Who was it that the letter was written to, and is there a copy somewhere?

Do you mean Darwin's letters to fellow geologist John Phillips? These are stored in the geological collections of Oxford University Museum, but, thankfully, they have all been transcribed and are available online as part of a project by the Faculty of Geology.
Phillips was a contemporary of Darwin's and was from Yorkshire, which may be why you have in your mind that the letters are kept in a Yorkshire museum.
In this letter (https://www.darwinproject.ac.uk/letter/?docId=letters/DCP-LETT-2521.xml;query=;brand=default) Darwin tells Phillips that he will be sending him a copy of the Origin of Species and asks Phillips to please read the whole thing straight through--even though Darwin is fairly sure that Phillips will be "dead against it." He tells Phillips that he need not send any reply to the letter but asks him to please think before he starts to generate any "awful anathemas" against what Darwin has said.
We do not actually have the letter that Phillips wrote as a response, but Darwin later said that the response was "cautious but decidedly hostile."
You can see some other letters written by Darwin to Phillips here: https://www.darwinproject.ac.uk/search?search-correspondent="Phillips,%20John";f1-author=Darwin,%20C.%20R.
I think the first one linked is the one you are looking for, and Darwin's debate with Phillips is one of the most famous debates. Darwin later read Phillips's own book and noted that he was not surprised, but disappointed, to find that Phillips had rejected Darwin's conclusions.