Writers' Forum magazine run a monthly writing competition - rolling entry, so you just submit whenever you like and they put it into the next competition. £3 for subscribes or £6 otherwise, but a £300 first prize every month, so worth a punt!
Off the top of my head, other reputable short story comps include Bristol, Bath, Exeter, Yeovil, Wells, Frome, London Magazine, Fish (expensive), Bridport, Mslexia, Words with Jam, Costa, BBC and Times EFG (you have to be published for those two), HE Bates, Brighton, HISSAC, theshortstory.co.uk (another monthly one), Commonwealth, Manchester. Have a look at Paul McVeigh's website for details of others.
For novels, there's the Lucy Cavendish, mentioned above. Also Bath, Exeter, Mslexia (not every year), Bridport, Yeovil.