they have a meter you have the fill up completely in order have a strong flame and then they'll give you the reward

You need to give them wood each day. It replaces the carrot button. The wood gives points to the meter. Water gives -1 point. The wood still comes from your inventory so you'll need a store of it to be able to give it to your Fire horses.

Does it show on the Horses page? As his flame never shows anything.

Thank you Eterna, I had no wood in reserve. Thank you!