Very locally there’s pleasant easy offroading around Sizewell Belts, and south towards Thorpeness, with some interesting (cheeky) bits along the cliffs. Slightly further afield there’s lots of bridleways up towards Dunwich forest, although you’ll need to be careful of your route around Dunwich Heath (National Trust) and Minsmere (RSPB) as these are sensitive nature reserves and best avoided.
As Johnny says, Tunstall forest is the place for a bit of proper singletrack – have a look on the forum and you’ll be made welcome on a ride. If you’re feeling keen it’s a rideable distance from Sizewell – my folks live a bit further up the coast and I’ve occasionally done a 90% offroad route down to Tunstall and back, can email you more details/GPS if you’re interested.