“A week off” sounds like “a wee cough.” “OUR DAILY SONG HIT: ‘I Have a Wee Cough, So I’ll Take a Week Off’” was printed in a 1924 newspaper. “Because of this, the Scotsman took a week off—A WEE COUGH” was printed in 1980 newspapers.
 
A joke was posted on Sickipedia on December 10, 2011:
   
“Phoned my work this morning and said, ‘Sorry boss, I can’t come in today, I have a wee cough.’ Boss says, ‘You have a wee cough?’  I said, ‘Really, Cheers boss, see you next week.’”
 
Similar to “wee cough” (“week off”) is “far cough” (“fuck off”). “People must not cough near you, they must cough far away…” is a related joke.
     
           
5 September 1924, Casper (WY) Daily Tribune, “Lines and Angles” by Ted Osborne, pg. 8, col. 4:
OUR DAILY SONG HIT:
“I Have a Wee Cough, So I’ll Take a Week Off.”
 
19 February 1980, Chicago (IL) Tribune, “Jumble” word game by Henri Arnold and Bob Lee, sec. 4, pg. 10, col. 5:
Because of this, the Scotsman took a week off—A WEE COUGH
       
Twitter
ScotEduBlogs
@ScotEduBlogs
ScotEduBlogs: A wee cough does wonders http://tinyurl.com/2945ef
4:08 PM - 21 Oct 2007
     
Google Books
A Man Walks Into a Bar 3
By Jonathan Swan
London, UK: Ebury Press
2010
Pg. 349:
Why is a Scottish boy with a cold like a soldier with seven days’ leave? Because they both have a wee cough.
 
Twitter
Dorkomatic
@dorkomatic
I said to my boss “I have a wee cough” - I think he misheard me.
4:08 AM - 24 Sep 2010
 
Twitter
Lee Isserow
@Lee_Isserow
Scottish accents are stupid. “A week off” sounds like “A wee cough”. Confusion ensued. Hilarity was, unfortunately, absent.\
9:36 AM - 13 Jan 2011
   
Twitter
Fresh Stand Up
@FreshStandUp
Phoned my work this morning and said:
” Sorry boss, I can’t come in today, I have a wee cough.”
... http://bit.ly/sBabmA  via @sickipedia
9:08 AM - 10 Dec 2011
 
Google Groups: uk.rec.humour
Sorry boss, I can’t come in today
Grr
12/10/11
Phoned my work this morning and said:
 
“Sorry boss, I can’t come in today, I have a wee cough.”
 
Boss says “You have a wee cough?”
 
I said “Really, Cheers boss, see you next week.”
 
18 December 2011, Sunday Sun (Newcastle-upon-Tyne, UK), “Jaffa’s Joke,” pg. 49:
I PHONED my work this morning and said: “Sorry boss, I can’t come in, I have a wee cough.”
 
The boss says “You have a wee cough?” I said: “Really? Cheers boss, see you next week!”
   
Google Books
Sounds Interesting:
Observations on English and General Phonetics
By J. C. Wells
Cambridge, UK: Cambridge University Press
2014
Pg. 74:
For some reason it reminded me of the Scotsman who took a week off work because he had a wee cough.
 
Reddit—Jokes
I called work this morning and whispered, “Sorry boss, I can’t come in today. I have a wee cough.” He exclaimed, “You have a wee cough!?”
submitted March 6, 2018 by madazzahatter
I said, “Really?! Thanks boss, see you next week!”