How much does oxygen enrichment improve some combustion engines?
What does &quot;energy efficienct&quot; mean?
It depends on what you mean by &quot;efficient&quot;.
For combustion reactions that involve engines, if &quot;efficiency&quot; means how much work (the &quot;good&quot; type of energy) that comes out of the engine per unit input of fuel, then yes oxygen-enriched mixtures would be more efficient than regular old air.
If instead &quot;efficiency&quot; means how much of the fuel is combusted all the way to $\ce{CO2}$, the answer is also probably yes, but for different reasons.
The efficiency of engines
To understand why, keep in mind that no engine can ever be more efficient than an ideal Carnot engine.  The efficiency of a Carnot engine is $\frac{T_H - T_C}{T_H} = 1 - \frac{T_C}{T_H}$, where $T_C$ is the temperature of a cold reservoir (e.g. our everyday environment)   in Kelvins and $T_H$ is the temperature in Kelvins of a hot reservoir.
Thus, the hotter you can get the &quot;hot&quot; side of Carnot engine, the better.  In a car engine or other Otto engines, this would correspond to the maximum temperature of the combusted air (or oxygen)-fuel mixture just before the expansion stroke.
Adiabatic flame temperature
With air instead of pure oxygen as the oxidizer, some of the energy of combustion goes into heating up inert components of the air, mostly nitrogen.  This lowers the maximum temperature of the air-fuel mixture and decreases theoretically attainable efficiency.
The adiabatic flame temperature is a quantitative number behind this idea.  Wikipedia lists the adiabatic flame temperature, the maximum temperature attainable in the combustion of certain fuels, and the values for fuels burning in air are a bit less than the same fuels burning in oxygen.
Limits of this analysis
Of course, there are many factors that make real engine performance different than ideal engine performance.  In fact, designing engines that can operate with at the extremely high temperatures that would maximize efficiency is tough because of materials problems.  (The adiabatic flame temperature of gasoline in pure oxygen is somewhere around 3800 K, far higher than the melting temperature of steel, for example.). So whether a given real engine designed to work under particular sets of conditions will necessarily do better with pure oxygen instead of air is less clear.  Probably yes?  And for sure an engine designed appropriately for use with pure oxygen would be more efficient than an equivalent one designed for us with air.