Review the original item:
Professor Britton was grading papers at her desk,
her students were slaving over their in-class compositions, sighing
heavily and flipping through their dictionaries for the correct spellings
of words.
You wanted to fix it this way:
Professor Britton was grading papers at her desk her
students were slaving over their in-class compositions, sighing heavily
and flipping through their dictionaries for the correct spellings of words.
To delete the comma between desk and her would cause an equally bad problem, a fused sentence. A fused sentence occurs when you have two main clauses joined with no punctuation whatsoever. Professor Britton was grading papers at her desk is the first main clause. Her students were slaving over their in-class compositions is the second main clause.
The spot between desk and her needs a stronger break than blank space.
You might want to consult the rules for fixing comma splices and fused sentences.