Sometimes gestured with a motion of slicing the neck with your finger, this means "to get fired" or "to get sacked" from a job. This originates from when decapitation was a form of punishment, like in the Edo period, for instance.
Here is another "neck" related expression, which means to "be deep in debt". Imagine someone who is so deep in debt as if it were physically all around them that they cannot turn their head at all (by rotating the neck) to look around. It's not funny to be deep in debt, but this expression is.
From the full expression "下手の横好き", this means to love doing something but at the same time be terrible at it. "横", in addition to meaning "horizontal" or "sideways", can be used in other expressions like 横車を押す (to have one's way against all reason) or 横紙破り (act illogically), and carry the meaning of something like "misdirected/wrong" or "against logic". The word 横好き isn't really used outside of the expression 下手の横好き, so just using 横好き will trigger the meaning.
Besides it's literal meaning of roasted mochi, this means to be jealous. There's a "good" word for jealous in Japanese: "羨ましい（うらやましい）" which is more like envious, and then there's "焼き餅". There's another word for jealousy too, 嫉妬（しっと）、and it's said that the word 焼き餅 was first created from 妬く気持ち or やくきもち literally meaning "jealous feelings" (which uses the second kanji in 嫉妬 for the word jealous in 妬く）and became a play on words to say 焼き餅 and eventually came to mean "jealousy". There's also another thought that when jealous, one's face takes on a sulky look resembling roasting mochi (which expands and gets gooey). I had actually always thought it was because when you see people roasting mochi you get jealous of eating it. The full expression is "焼餅を焼く".