Your draft is your essay. Any writing can be improved but you have to know when to stop. The hand-in date is the absolute limiting factor, but if you plan your essay carefully, you will leave yourself time to go through your draft to check you have covered all the criteria, structured your essay well, checked your grammar, spelling and punctuation. Far too often for students, the first draft is also the final draft. This shows poor planning. Write your whole essay but leave time to go through it and improve it.
The tutor who sets your essay will probably want to see you at an early stage. This could be with your outline or when you have started your first draft. At postgraduate level it might even be when you have decided on a title.
You might make a specific request to see your tutor. Perhaps you think you do not have enough ideas, or perhaps you need some advice on what to read. Whatever the reason, it is essential you go to the tutorial well prepared. This means you should have thought through what questions to ask. Too many students arrange a tutorial and then waste the time (which may be as little as 15 minutes) by expecting the tutor to do all the work.
If you have arranged a tutorial, it makes sense and it is only polite to let your tutor have a copy of what you have written in advance so that he or she can read it through beforehand.