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you can run them again, but once you get the championship prize you can not get it again.. you can even click on it and "give up this championship" it will erase your results, but doing the races again still only gets you the 2,000 per race, no big bonus..
do some missions if you need money, look especially for the convoy ones.. don't understand why they are called convoy, they are delivery missions really, bring their car to the shop. i've had 2 convoy missions for 50,000 and one for 65,000. also lets you drive some nice cars, gain experience, and can find new buildings since you usually bring them to a car dealer or tuner shop.
i tried being serious once, in 1989.. didn't particularly care for it.. so don't expect it to happen again.