I know this is an older thread, but maybe this will help. A cigarette lighter is NOT meant to constantly supply power to the socket. It's purpose is to heat the element in the lighter and then shut down. The problem is probably in the wiring to the lighter or the fuse for the lighter. A lot of people have pulled their hair out trying to track these issues down over the past several years. I'd suggest using the lighter just for lighting your smoking choice and the power ports for charging phones or other usb items. If you need additional items, add a usb splitter.
