Writing Contest list of fiction
There is a "Contests & Awards" section in the Writer's Market book that is published every year. There are several essay contests in the Non-Fiction category in that section.
For a much bigger list, you can do a Google search for essay contest or the like. Be sure to click "More Search Tools" and choose to limit the search to within the Last Year so that you aren't wasting your time on old, out of date, contests.
If you have a specific arena you are interested in, the best thing is to find the essays through professional groups in that field. For example, if you wanted to write essays about law, the Bar Association website lists ABA Essay and Writing Competitions
While I'm sure you will find plenty of websites out there with lists of contents, none of them will be exhaustive and many will be out of date. You'll just have to go fishing :)
One word of caution: Charging a fee for entries to writing contests is a very lucrative business. Make sure that the contest is worthy of your entry. The more well-known and reputable the organization running the contest, the better.
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Ten-year-old from Harpenden wins prize in a national writing contest — St Albans & Harpenden Review
A ten-year-old from Harpenden has won a prize in a national contest designed to find the best young writers. Susanna Tredinnick said she had to pinch herself when she was announced winner at the Wicked Young Writers' Award held in London.