If the awakening fails, the training percentage (%) will be reduced back to 0% and you will need to train the horse again. You wil gain +1 Awakening failstack, which increases your chance of success on your next attempt.
The formular for working out your chance of success is (1% + (o.2% * failstacks)). This means that every failed attempt increases your chance by 0.2%.
You can also choose to use 100 Cron Stones when awakening. Using Cron Stones will reduce the training percentage to 100% instead if it fails.