Here are three such contests, all of which require poems written to fit a certain theme.
Inglis House Poetry Contest
The consistent theme from year to year is disability The contest used to be restricted to disabled poets but recently added an open category for all poets. With the two-category system, poems may now reflect first hand experiences of the disabled and/or an understanding of the issues surrounding disability.
Annual deadline: June 1st
Prizes (in each category): $50, $30, $25 plus anthology publication
Don't let the title fool you. Just because you're not a Scot living in London doesn't mean you can't enter. But you do have use that premise as your theme. And bear in mind that the dinner portion of the prize takes place in London, England. Guess how you'll be spending your prize money?
Deadline: August 31, 2011
Prize: £1000 plus two tickets to ScotsCare's St Andrew's Dinner; the top five poems will be displayed in ScotsCare's offices and published in ScotsCare Shout.
Family relationships are at the heart of this competition. Poems should be emphatic in nature and reflect personal experiences with relatives, both good and bad. The experiences, not the relatives--or maybe not. ;-)
Deadline: December 31, 2011
Prize: £100 plus publication in online magazine
By posting these contests, I am in no way endorsing them. Exercise due diligence and check them out before participating. And be sure to follow the guidelines if you do decide to enter. Good luck!