I now have a full version of the caricature of Lovecraft’s New York friend and correspondent Ernest La Touche Hancock (1857-1926). It was only available previously as a tiny thumbnail, back in 2013.
The name on the donkey appears to be “Pegasus” (the immortal winged horse of myth), and the lettering on the tiny toon tableau in the bottom-right corner cannot be read. The figure in the Union Flag waistcoat is “John Bull”, the archetypal beef-fed 18th century British squire. Hancock wears a small ‘mortar board’ hat, which once symbolised a teacher. The only thing that can be fathomed today is “John Bull” — like Lovecraft, Hancock was an ardent Anglophile.
Hancock was familiar with many cartoonists of the 1890-1925 period and his long survey article “The American Comic and Caricature Art” (the American The Bookman, Nov 1902), he praised the young Herriman of Krazy Kat fame: “Art combined with poetry is the characteristic of George Herriman. Were his drawings not so well known one would think he had mistaken his vocation.” It’s thus not impossible that Hancock, knowing of Lovecraft’s liking for cats, might have mentioned the poetick Kat in a letter.