Can I ever reattempt a horse race after placing lower than first place?
When horse races first become available, there are two races available. Both are bronze league, one easy, and one difficult. I placed second in the difficult race. The NPC won't let me try again, stating the race is not available.
Can a race ever be reattempted, to try and get the first place reward?
Sure can! You need to come back, the races are only available once when you enter the town of Gallopolis for the first time. Once you enter the city later in the story, there are multiple cups to try and you can enter as many times as you like.