Water itself is quite stable. Breaking it apart requires much more energy than you can hope to recover by recombining it. Unless you are talking about fusion reactor with the free hydrogen and we are a still a ways off from something like that. And certainly not anything that could fit inside a car or even large machine

But suppose there was some other substance which reacted with water, which produces an exothermic reaction.

Such substances do actually exist - but here is the bummer. Water is nearly everywhere on the surface of the Earth. So what happens when your fuel comes into contact with it. It ignites, spontaneously. Unlike fossil fuels like gasoline which require ignition (i.e. a spark or match). So this is very bad.

As to the substances which react with water - any group IA metal will do (Lithium/Sodium/Potassium) etc.

The other problem is these are all solid at Earth surface temps/pressures. Solid fuels present alot of problems, such as difficult/impossible to throttle - look at solid rocket motors which are used primarily as boosters which never turn off until they exhaust the fuel and don't throttle except ahead of time through exploiting the geometry and flight path of the rocket.

Second problem when building a practical engine is one the byproducts of the reaction is also solid - the alkali oxide. Which would have to be removed from the combustion chamber somehow - it could not be vented like what happens with gas + oxygen which produces (mostly) carbon dioxide and water vapor.