The word "hatchet" is correct and usable in written English.
You can use it as a noun meaning a short axe with a sharp blade at one end and a pointed tip at the other. For example: "I used a hatchet to chop the wood into smaller pieces."
Dictionary
hatchet
noun
A small light axe with a short handle; a tomahawk.
synonyms
Exact(54)
Lord Adonis, the former Labour transport minister, said the speech amounted to "a systematic hatchet job" on the culture at Network Rail, adding: "I don't think I've ever heard a CEO be so critical of his own organisation".
But Warsi's critics do a hatchet job on her: she was an inexperienced political lightweight, one who had failed to win winnable Dewsbury in 2005, failed to get another nomination before 2010 despite being on Cameron's A list, as a self-made, working class, northern, female Asian Muslim – that ticks a lot of boxes.
He describes this as a "hatchet job" as he was "was never editor-in-chief of the Sunday Mirror".
Men's 400m hurdles, final, 8.45pm Greene and Jackson insist they have buried the hatchet but there is plenty more noisy rivalry on Monday.
The two sides buried the hatchet, with Channel 4 subsequently announcing a co-production deal with Group M Entertainment.
You can read the No 10-guided indictment ("Warsi flounces out") in Wednesday's Mail coverage (relegated to page 6), including a notably bitchy hatchet job on "Baroness Blunder" by Andrew Pierce, who adds overambition and vanity to the charge sheet.
Similar(6)
"Even your boss, strait-laced, hatchet-faced Mr George Lowery, why even he can smell the rancid, pungent scent of sex all over you," he barks at her during one scene.
A twist or two of Basel How to spend it Bye-bye EMBI Carry on speculating Reprints Related items Correction: Japan's personal spendingMar 8th 2007 Economics focus: Carry on speculatingFeb 22nd 2007 Japan's currency: Carry on living dangerouslyFeb 8th 2007As it was, Mr Abe's hatchet-men had to be restrained from the same heavy-handed methods that prevented a rate rise last time.
Less conspicuously, he has been taking advice from Alastair Campbell, Tony Blair's hatchet-man.
Virtually everyone who has seen the painting of the hatchet-faced farmer holding his pitchfork that hangs in the Art Institute of Chicago (or the altered version that appears in the opening sequence of "Desperate Housewives") will recognise it as a famous picture.
When he worked as a tabloid journalist, he propagated the famous legend that John Major, then Tory prime minister, tucked his shirt into his pants; and "The Blair Years", the diary of his time as media hatchet-man to Mr Major's successor, is full of them.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com