If so, how can you do this? I just had a horse that, on race day I noticed had an issue in the paddock and I wanted to then drop them out, but couldn't. I'm forced to race the horse and now it's injured.

Save before each race, then when on the race screen before you go to the course you can save and quit the game using the usual buttons, then when you reload the game it'll be the start of the game and you can remove them from races. This is the only way I have found to do it quickly and easily for when they are sweating up or not fit to race but I didn't realize until we're at the race.