It's a detail from a cartoon by Charles Townshend, one of many he made recklessly ridiculing contemporaries. Townshend was a military officer and politician. He served under Wolfe at Quebec and later, as an MP for Norfolk, was responsible for the infamous Townshend Acts which he intended as just payment by the American colonies to help cover costs of their defense and which the Americans attacked as taxation without representation (first use of that phrase).
Sandwich is the subject of a good biography by N.A.M. Rodger. Rodger defends Sandwich against the accusations of libertinism and compulsive gambling made by contemporaries, but Sandwich did have many mistresses and was a famous gambler. He was also attacked as hypocrite for a speech in the House of Lords in which he effectively destroyed the political career of the notorious John Wilkes, a friend of Charles Townshend. (Sandwich showed Wilkes to be obscene and misogynist and Sandwich's enemies said he was no better; Rodger denies this.) Sandwich was also a naval reformer and administrative reformer in general. When Cook discovered the Hawaiian islands he named them for his patron, Sandwich and, it's true, the sandwich we consume many lunchtimes is also named for him. My favorite Sandwich connection is musical: He loved music, played it himself, and sponsored musical events at his home. He was passionately fond of Handel and took as lifelong companion Martha Ray, a young tradeswoman whose ability as a singer he particularly admired. (This alliance has its own separate story.)
In the Townshend cartoon Sandwich is made to say "I love deep play; this or nothing!" He has a cricket bat over his shoulder in place of musket. Deep play refers to a batting style in cricket but also underhandedness in politics -- an opportunism that was seen at the time as dishonorable, though common. I think the stuff dangling from the end of the bat are his cricket shoes and socks.
The cartoon was an immediate success -- outraging Fox and adherents and tittilating the political gossips in London coffee houses. It was immediately pirated; cards were printed showing each of the individuals it ridicules.
Here's the original. You see Fox leading a bunch of ill-assorted recruits towards an altar, on which is placed the king's son and commander-in-chief, the fat Duke of Cumberland, crowned with laurel. In addition to Sandwich, the recruits included Winchelsea (First Lord of the Admiralty), and Bubb Dodington.
You can read more about this in The Caricature History of the Georges.
Here is a cartoon showing Sandwich's first encounter with Martha Ray.
My source for the Sandwich caricature at top is the book, England Under the House of Hanover. The National Portrait Gallery has the pen and ink original:
John Montagu, 4th Earl of Sandwich
by George Townshend, 4th Viscount and 1st Marquess Townshend
pen and ink, 1751-1758
5 1/4 in. x 3 1/8 in. (134 mm x 80 mm)
Bequeathed by Robert Wyndham Ketton-Cremer, 1971