Learn English – Where does the word “totient” come from

Hack job has an interesting history. The sense of hack in play here probably originates with the oldest uses of the word as meaning "to cut irregularly or inexpertly." That usage dates back to Old English haccian, and thence to the mists of antiquity. It's not hard to see how the other senses of hack, many of which carry connotations of poor quality or amateurishness, would have emerged out of this definition.

Hack job begins to appear in the literature in the 19th and 20th centuries. Most of the early uses are from the 1920s or later, although I find one fascinating example in a book review published in 1837:

There is a freshness and talent about this unpresuming production which has exceedingly captivated our fancy. There is not a hack line in it : what a treat for a reviewer ! and a reviewer in these days, when one hack job succeeds another with such unremitting activity, that it almost seems, at the end of the season, as if we had read only one huge hack work.

As with most of the other early uses of hack job that I have found, the sense of hack being used here is that of the hack writer, who produces large quantities of mediocre writing because of the money it brings in, rather than because of inspiration or talent. Interestingly, this sense of hack is apparently completely unrelated to the ancient cutting/chopping definition: the OED relates it to an obsolete use of hack to mean a horse that draws a hackney cab, another form of drudge work. (Hackneyed, meaning trite or clichéd, is also related.)

Throughout the 20th century, the use of hack job to mean mediocre written work predominates. It takes on an interesting sub-definition around the middle of the century, when I begin to see the hack job label applied specifically to works that (in the opinion of the writer) constitute severe and sustained attacks on a person, belief, etc.--what we might otherwise call a hit piece. This usage pretty clearly borrows from the older, unrelated use of hack as meaning clumsy, violent cutting: the hack writer is hacking away at his target.

It is not until the 1990s that I begin to see hack job used to refer generally to quick, shoddy work in contexts that have nothing to do with writing or the creative process at all:

The Chinese sometimes cut a chicken into chunks, bones and all, before cooking it. It's a true hack job, like Kentucky Fried Chicken, but remember that the bones add to the flavor.

The broadening of hack job in recent years has no doubt been influenced by the senses of hack popular in computing, meaning variously "to break into a computer system" and "to write computer code for pleasure, or to derive pleasure from writing computer code." This is supported by the appearance around the same time of the phrase hack together (roughly meaning "to create or assemble quickly or inexpertly"), which was initially used only as hacker jargon but is today, I'm sure we would all agree, understood by a wider audience.

That hack job came to exist as an idiom was probably inevitable: woodcutting is a job, being a hack writer is a job, so it's hardly surprising that the term would arise eventually, and in fact it has probably been coined multiple times independently. The evolving use of hack over the years has unsurprisingly contributed to an evolution in the way we use hack job as well.

Nigel Rees actually published a book titled More Tea Vicar? An Embarrassment of Domestic Catchphrases (2009), but its entry for that phrase is disappointingly vague:

more tea, Vicar? A correspondent who, understandably, wished to remain anonymous advanced the family phrase, 'for after a fart, or to cover any kind of embarrassment'. Paul Beale has collected various forms for a revision of Partridge/Catch Phrases, including 'good evening, vicar!'; 'no swearing please, vicar' (said facetiously to introduce a note of the mock highbrow into a conversation full of expletives); 'another cucumber sandwich, vicar' (after an involuntary belch); 'speak up, Padre!/Brown/Ginger (you're through)' (as a response to a fart).

Rees (again) in A Word in Your Shell-like (2004) provides additional information on the phrase:

more tea, Vicar? Phrase for use after a fart or to cover any kind of embarrassment. British use, from the 1920s/30s? ... David Rogers declares: 'The phrase, "More tea, Vicar?" has entered the language as shorthand for comfortable suburbia.' Hence these stories: '"More tea, Vicar?" asked Lady Lavinia as she poured the tea with her other hand' and 'One day the young Vicar was visiting two elderly ladies. Whilst he was sitting on the shiny sofa, he passed wind mightily and noisily. As the echoes died away, one of the ladies filled the embarrassing silence by asking, "More tea, Vicar?" "Oh no!" he replied," "It makes me fart!"'

On the other hand, Julia Cresswell, The Cat's Pyjamas: The Penguin Book of Clichés (2000) indicates that "More tea, vicar?" was a catchphrase associated with sedate gentility before it got commandeered by jokesters:

The tea party expression [which arises in connection with "behaviour that would make something less outrageous look like a vicarage tea party"], and the associated catchphrase More tea, vicar?, have been in use as a comparative standard of innocence since at least the 1950s.

In any case, a Google Books search doesn't turn up any matches at all for "more tea, vicar" before 1981, when the phrase appears in New Zealand Alpine Journal, volume 34, not in the context of farting or complacent gentility but of a surprisingly easy ascent:

Well, the photos we took from the air must have been tilted, because we wandered up in perfect conditions as though it was a Sunday picnic. More tea, Vicar!

Under the circumstances, I think that the circa 1985 date put forward by The Concise New Partridge Dictionary of Slang and Unconventional English (2015), noted in Josh61's answer, is far closer to the mark than Nigel Rees's "from the 1920s/30s?" as the starting point for saying "More tea, vicar?" after someone farts.