Just wanted to mention these as I know Year 13 and 13 are busy and the closing dates for lots of essay competitions are in May / June if not before -
John Locke Essay Competition - good for anyone applying for Politics / Economics / Law / History / Philosophy / Psychology / Theology as there are separate categories for each. Worth a go as three prizes in each category and lots of commendations - plus the chance to go to Oxford for the prize giving.
Dukes Essay Competition - they have to write a short essay in response to a one-word title. They can apply their own subject interest to the title. Any subject.
They can do a search for subject magazines at Oxford / Cambridge etc and maybe see if there are any submission requests for short articles etc.
For Geography / Anthropology / Earth Sciences / Geology / Sociology etc there is Routes Journal where they will help students to get an essay / article / book review actually published.
Financial Tomes run an essay competition for Economics (I think)?
Anyway, just some ideas in case anyone is interested. There will be other competitions so worth a Google. Even if they don't win, it helps with the PS to have developed an area of research.