Applying for essay competitions?

I am really considering applying to Cambridge next year, for either history or land economy. I am trying to decide whether to enter two essay competitions which have their deadlines in July. One is for history run by a college at Cambridge, and the other is the young economist of the year competition. I realise I have left it a bit late, but was just wondering if it is worth me entering. I am prepared to put a lot of time into it over the next few weeks but understand they are highly competitive and wonder if there is any point if I don't have a chance of winning. I am targeted As and received A in history and A* in economics in my mocks, but I am worried these competitions are always dominated by private school students with incredible access to resources and therefore I would stand no chance. Would I be better off putting my time into something else to help my uni application? Thanks