A wretch is an unhappy or unfortunate person, especially one in the depths of misery of some sort. The word has several senses extending from this one; it sometimes refers to a person who is despicable or contemptible but not necessarily unfortunate, and it’s sometimes used for animals. It’s also the source of the adjective wretched, meaning miserable, unfortunate, or (describing nonhuman things) very bad.
To retch is (1) to vomit; (2) to try, voluntarily or involuntarily, to vomit; or (3) the clear the throat by coughing or hawking. The word is primarily a verb, but it also works as a noun for the act of retching.
The two words are unrelated and have never been variants of each other, but, as is often the case with pairs of homophones that are both somewhat rare, retch and wretch are commonly mixed up. That spell check has nothing to say about phrases like “I wretched up my dinner” and “he is a retched creature” also doesn’t help.
The sight of the government now having to back Andrew, as he is “unsackable”, makes one want to retch. [Guardian]
The rocking of the deck beneath his feet made his stomach heave, and the wretched food tasted even worse when retched back up. [A Dance with Dragons, George R. R. Martin]
Even the hint of fish sauce or anchovy paste causes instant retching and gagging. [Huffington Post]
Field thought it was appropriate to get a bloke to paint the house in return for sorting out the poor wretch’s immigration status. [New Zealand Herald]
Unless he was royalty, which all too plainly he was not, my mother would have berated or possibly beaten this poor wretch. [Martin's Absolution, M. Clement Hall]
I gathered, from the opened food containers and dry water bottles, that the poor wretches must have been trapped there for some time [Daily Beast]