Is it possible to have a cyst removed from my ovary under local anaesthetic? as i am pregnant and don't want to take the risk of general anaesthetic

No,: Your ovaries are internal and require an incision. Local anesthesia would work for the skin incision but once inside pain is spread over a larger area. You could have a spinal or epidural which involves local anesthetic to the nerves in your back, but there is some systemic absorption. But if it is done with a laparoscopy general anesthesia will probably be required.