Green’s Dictionary of Slang

(US) tiny, very little.

1943		Wolfert Und. 415: Just a weenchy [...] little dash of perfume [W&F].
1960, 1975		Wentworth & Flexner DAS 571/1: [...] Dial. use since c1900.
2002		S. Bowman ‘Salinger’s Problem with Westerners’ posting 23 Oct. at RoughDraft.org 🌐 Yocum & Patrick sound a good deal more serious about books than I ever was. But to consign the three of us, & our like, implicitly, to the ranks of the knuckle-trailers is a teenshy, TEENSHY, weenchy bit – dare I say it ? – arrogant.