To get a big head start, accept the mission then respawn the vehicle and boost away, it won't fail the "ready,set,go objectives and give you a few seconds. Also don't worry about killing anything, just focus on the race. Just boost as much as possible and try to take the ramp near the end of the lap, good luck :)
The relic definitely does help, I know since I tried with it on. But you don't necessarily need it to beat the race. This video shows what route you need to take. Not my video, but just wanted to share it. Hope this helps, man!