No cure. You need to find what's causing it and treat that. This time of year, that pukey morning feeling is pretty common, especially in areas that have cold weather. With the warm/cool/wet fall we've had, my sinuses are going crazy too. Having that nasty stuff dripping down your throat into an empty stomach all night tends to make me really nauseous when I wake up - maybe that's what you're feeling too? Try a decongestant at night?
Otherwise, my pregnancy-morning-sickness cure was to keep some oyster crackers by the side of the bed. Before I got up, I'd chew on a couple of them and that helped a lot.