Yes, the very same Thomas “Rule Brittania” Arne. He also luckily wrote a number of catches. Born, spending most of his career and dying in and around Covent Garden and Drury Lane, it is unsurprising that he was most associated with the theatre, writing incidental music and opera (one of which supposedly influenced another naughty person, Mozart).

This song is rooted in the operatic world Arne was a part of and describes a little domestic affair. It refers to a lady (the drawing room jest being probably what one today would call a flirt or a Jezebel) who has (as was common in those days) fallen in love with a castrato singer named Senor Caponi (the name back then often used to innocently mean “Bighead”, but also would have suggested the more graphic connotation to the capon or castrated cockerel).

Another fellow, Sir Humphrey Split Whisker (a whisker-splitter being a term used for one given to sexual intrigue), is besotted with Fanny, but also clearly frustrated and angry with whom he considers a 'half-man" receiving her attentions and not him so has a bit of a hissy-fit at her father. 

Poor Señor Caponi (and his anatomy) is described with a rather large amount of derision and pity. In reality, castrati were generally very successful, lauded and wealthy celebrities, so whether or not intentional, one suspects a clear air of jealousy here and perhaps a perceived threat to his masculinity. The earlier mention of no treasures within suggests that Sir Humphrey's later offer of 'stoning her to death' may well be euphemistic. Who'd have thought?

 

“Bewitch’d Lady Fanny, the drawing room jest, lov’d Senor Caponi, a half-man at best;

What a Game play’d the Fool, when all she could win, was a poor shrivel’d Bag, and no Treasure within.

Sir Humphry Split Whisker, stark mad for the Girl, this Sentence pronounc’d to her Father, the Earl;

She ought, like Saint Stephen, to yield up her Breath; assign her to me, and I’ll stone her to Death.’’”