There used to be a competition by TCM for young composers - but that seems to have disappeared.
I can't recommend (or know of) any major film score competitions but here's my advice if you want to meet people and network.
Go to film festivals - there are SO MANY, and many of them have lots of up-and-coming filmmakers for you to meet. This is the best and most social place to meet filmmakers in my opinion. Everyone just wants to have a good time, so it's easy to make friends.
Go to a film school and advertise your services. You might not make much money, but you'll meet filmmakers. From your flag it says you're based in the UK, so you could head to the National Film and Television School for example. There are lots of filmmakers in the UK,
Apply for Film Scoring Workshops.
I HIGHLY recommend this - especially the first I'm going to recommend, which is free if your work/application is accepted (you do have to play travel and accommodation)
Richard Bellis - who runs this workshop - is an amazing mind. If you haven't read his book 'The Emerging Film Composer' then you must. It is in my opinion a mandatory read for any composer thinking of trying to break into the business - because it gives you a no nonsense approach to every aspect of the industry.
Back to his workshop - I know 4-5 composers who have been selected for this workshop and they all had a fabulous time. Apart from the massive benefit of being able to record with some of LA's finest, you learn an incredible amount in a short period of time. Competition is fierce, but if you're trying to break into the industry - it is ideal.
Here is another one from ASCAP but it's not free:
There are also summer schools as well, particularly in the US at places like Berklee, UCLA, etc... but don't pay for the stuff unless you're really committed to it (that's why I recommend the ASCAP workshop first, because it's a free workshop)
Beyond that, Google Search is your friend, but...film scoring competitions are in general few and far between. It would be far more worth your while to score some short films and get your music screened at festivals and out into the public.
Hope that advice helps :). All of these methods require a bit more leg-work, but it's actually really easy to make contacts and friends when you're not sitting behind a computer!