This is twoth in the sense of 'second', rhyming with tooth; not the combination of twee and Goth, rhyming with moth.
A correspondent writes to say she heard someone say 'a thirty-twoth of an inch' rather than the expected 'thirty-second', and wonders whether I know this usage, as it's the first time she's encountered it.
As a playful alternative to second it's been around a while, though the OED has no file on it. I can recall playing with this word as a child, and saying such things as twenty-oneth and twenty-twoth. Certainly, the force of analogy from the usual -th ending has had an impact on the irregular first, second, and third, so we do find oneth and threeth as well.
There's plenty of playfulness online. I've just done a quick search and found chapter the twoth, the twoth of July, part the twoth, and many more, as well as quite a wide range of jocular expressions relying upon it. One person bought a second toothbrush and called it a twothbrush. US columnist Larry Levy has a piece on two-player games headed The twoth, the whole twoth, and nothing but the twoth.
The word has a minimalist entry in Wiktionary, so plainly it is widely recognized. It's labelled 'nonstandard', and that's correct. But is there any evidence of a usage in standard English? Yes, in mathematics and computing, where such forms as nth and zeroth are also found. We find such expressions as 'the twoth-complement number'. I don't think this use of twoth is especially frequent, but it exists.