Thank you, CalifJim, Khoff and Eimai_Anglos.
I think I made a mistake -- yes, it should be a "gerund" and not "genitive."
I did some reseach on my own and I think I have gotten the following Google search results:
"a cutting of" 29,200 hits
"a shaking of" 29,700 hits
"a mixing of" 275,500 hits
All seemed to fit somewhat very well with the notion of an instance of cutting, shaking and mixing (as taught us to) by CalifJim, I think).
As to the original phrase, I think the following results were the product of a Google search:
"a barking of" 1,540 hits -- some entries dealt with a myth or a folklore.
"barkings of" 15,000 hits -- the entries here seemed to be from somewhat solid sources where the quality of writing leaves little doubt as to their acceptability, in my opinion, and these many entries from seemingly "quality" sources cast some doubt as to the credibility of what I believe to be Eimai_Anglos' assertion that the word "barkings" is not a correct English word (I think that is what he said).
Here, I am trying my best to get a grasp of what seems to be an important thing to know. Can you help? Can a gerund function as a countable noun? If so, can you show me how it is done in simple terms?
Thank you, all.