There is no such word as "averagely"! You would need to express the thought in a different way ("adequately written," for example, but you could come up with many solutions). "So-so" is a good word to avoid. It sounds old-fashioned, and it is never used before a noun.
This is so- so written essay. No article and 'so-so' doesn't work as adverb or adjective.
'So-so' is used after a verb and is informal. 'This essay is only so-so'. 'The quality of this fabric looks so-so'
'Mediocre' is a good word. 'This essay is a mediocre standard'. 'The writing in this essay is mediocre'
I can't think of any word to put before written.
This is badly written essay.
No article. I am not sure if 'badly written' needs a hyphen or not. A good word option for bad/badly is poor/ poorly.
I am not sure if you need 'written' each time.
The writing in the essay has to do with the sentence structure, grammar, rhetoric, style... If you just say "the essay", you could be referring to the topic, the arguments presented, the evidence offered, the conclusion drawn...
I think spell-check and Webster's serve their purpose for documenting whether a word is out there and whether it fits the basic criteria of "standard" usage. But at a certain point common sense and esthetics have to play a role as well, or else we will end up speaking and writing like Google translations!