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The phrase clergyman’s sore throat designates chronic pharyngitis.

This phrase refers to preachers who overstrain their voice. The following, for example, is from Dysphonia Clericorum, or Clergyman’s Sore-Throat: Its Pathology, Treatment, and Prevention (London: Longman, Brown, Green, and Longmans, 1848), by James Mackness, M.D. [Introduction: page 3]:

Never was there so great a need or demand for preachers: population rapidly increases, new churches are built and thronged with dense masses of hearers, and for every thousand within the walls of a sacred edifice there are tens of thousands without. A natural result of this state of things is, that the most zealous and valuable of the ministers of religion feeling most deeply the necessity of exertion are led to make efforts of an exhausting and destructive nature to meet the wants of their fellow-countrymen; and hence the prevalence of that peculiar form of disease which is the theme of these pages.

—Cf. also the expression tennis elbow.

With one exception, all the earliest occurrences of the phrase clergyman’s sore throat that I have found are from medical texts published in Philadelphia, Pennsylvania, USA—these early occurrences are as follows, in chronological order:

1.1 & 1.2-: From an article published in The Eclectic Journal of Medicine. Edited by John Bell, M.D. (Philadelphia, Pennsylvania, USA) of December 1837 [Vol. 2, No. 2]:

1.1-: [Title of the article: page 66]:

LARYNGEAL PHTHISIS.—CONSUMPTION OF THE THROAT.—CLERGYMAN’S SORE THROAT.

1.2-: [Closing paragraph of the article: page 76]:

It was our intention, at the beginning of this article, to close it with some remarks of a hygienic character, respecting the causes which operate with such direful force and frequency on gentlemen of the clerical profession, in the production of laryngeal phthisis. But we find that to carry out our original plan would lead to a more lengthened essay than is compatible with room and the introduction of other matters into the present number of the Journal. We shall, therefore, postpone until next month what we have to say, in the shape of retrospective notice, existing errors, and an amended future, bearing on the subject of Clergyman’s Sore Throat *.

* However, the phrase clergyman’s sore throat did not reoccur in Volume 2 (from November 1837 to October 1838 inclusive) of The Eclectic Journal of Medicine.

2-: From Lectures on Clinical Medicine, delivered at the Philadelphia Medical Institute, by W. W. Gerhard, M.D., published in The Medical Examiner (Philadelphia, Pennsylvania, USA) of Wednesday 12^th September 1838 [Vol. 1, No. 19, page 301, column 1]:
—The following, from a lecture on the treatment of chronic laryngitis, is about the application of leeches:

When there is soreness of the larynx and trachea, and pain on pressure, without the permanent alteration of the voice and appearance of the sputa indicative of ulceration, you may take from two to six ounces of blood, with almost certain advantage. The soreness of the larynx which accompanies the clergyman’s sore throat, is nearly as certainly relieved as that which is confined to the peculiar affections of this organ.

3-: From Frightful State of the Clergy, published in The Medico-chirurgical Review, and Journal of Practical Medicine (London, England) of 1840 [Vol. 32, No. 63, page 284]:

The Church is in danger now. We would advise all parsons of the good old style to beware of their evangelical brothers. Better stick to the “jolly full bottle,” than join the snivelling tea-totallers [sic].
It appears that in New England the clergy are in such a strait, that they have set a new quinsy a going. This is called the “clergyman’s sore-throat,” and is distressingly prevalent.

4-: From John Bell’s twelfth lecture, on chronic laryngitis, published in Lectures on the Theory and Practice of Physic. By William Stokes, M.D. With […] Twelve Additional Lectures, by John Bell, M.D. (Philadelphia: Haswell, Barrington, and Haswell, 1840) [page 657]:

The disease which is the subject of the present lecture has been variously named. In addition to its technical designation of  Chronic Laryngitis; Laryngeal Phthisis; Laryngitis with Secretion of Pus; it has received the popular ones of Clergyman’s Sore Throat: Throat Consumption, &c.

5-: From Lectures on the diagnosis and treatment of diseases of the lungs, by W. W. Gerhard, M.D., published in the Medical Examiner (Philadelphia, Pennsylvania, USA) of Saturday 6^th February 1841 [Vol. 4, No. 6, page 88, column 1]:

The disease known by the name of chronic laryngitis, or sometimes “clergyman’s sore throat,” is occasionally connected with phthisis.