Carry not spread (?)
While there is some evidence that mosquitoes may "carry" lyme it is unproven that they can transmit it.
The best discussion have found on this subject is http://www.Rosemarydrisdelle.Com/?P=412
i have seen many reports claiming that lyme and bullseye rashes have been documented after mosquito bites but have not found references proving this. So to the best of our knowledge, the answer is no.