Yes, the exhaust can be fuel / useful source of energy. It contains lots of pressure and energy as heat.
You can for example:
- Use the exhaust heat to heat up the heating system of the car, speeding up engine and cabin heating
- Use the exhaust pressure to drive a turbocharger which compresses more air into the engine, meaning a smaller engine is sufficient, using less fuel
I'm sure you'll find many cars already use either or both of these options.
Now, for carbon monoxide and unburnt hydrocarbons: yes, they contain some energy, but very little as @alephzero found. Far more energy is in the pressure and heat of the exhaust.
Also, if you could somehow store the carbon dioxide, you could combine it with hydrogen produced by electrolysis using clean electricity (solar, wind, hydro, nuke) in water gas shift reaction to produce lots of carbon monoxide, which using Fischer-Tropsch and more hydrogen can be converted into liquid hydrocarbons.
Unfortunately, there is no practical way of storing the concentrated carbon dioxide in exhaust, and there is so little carbon dioxide in the atmosphere that capturing it for use with water-gas shift reaction and Fischer-Tropsch synthesis is expensive.