They both mean the same thing, but check over specifically has connotations of proofreading and finding any errors, and also of checking the whole thing.
Over in this sense cannot replace for, though. It's still needed. If I were to say this phrase, I would not split check and over (in this context), as that sounds jarring.
I would suggest:
I have to check over my essay.
With for mistakes being implied, as in this definition from Merriam-Webster:
Definition of check over : to look at (something or someone) in a
careful way to find problems, mistakes, etc.
checking the new cars over
The doctors checked him over for bruises.
Be sure to check over each item on the list for any mistakes.