kicking off or kicked off

You can use either, but your choice changes the sentence's meaning slightly. You should choose one or the other based upon what you want to mean.

I'll consider "kicked off" first.

The race is a long-distance one, kicked off with a special song ("Giin-Goo") that all the horses know.

"Kicked" is sometimes the past tense form of "kick", but here it's the passive form. If you want to emphasize the race as being driven by the participants (including organizers, spectators, and musicians), you would use "kicked".

So now, consider "kicking off".

The race is a long-distance one, kicking off with a special song ("Giin-Goo") that all the horses know.

"Kicking" is the progressive or "continuous" form. Here, there's no outside agency driving the action. So, if you want to emphasize the race as a (possibly living, autonomous) thing that drives itself, you should use "kicking".