Efficiency of the Kuznetsov NK-88 or NK-89 engine burning hydrogen
Based on my reading, the biggest disadvantage of hydrogen as an engine fuel is its low density.  Even when stored as a cryogenic liquid, it take up more than ten times the space as the same mass of kerosene, which means the same size tanks will hold only about a third as much energy (even though hydrogen has about three times the energy content per kilogram as kerosene).
Given these qualities, an engine suitably redesigned to burn hydrogen ought to be able to run more efficiently in terms of specific power output (energy produced per mass of fuel) or specific fuel consumption (same thing only upside down -- fuel consumed per energy produced), but even ignoring the weight of insulation and the fuel cost of boil-off in flight (or before takeoff), an aircraft the same size would have roundly a third of the range.  This is usually taken to mean that hydrogen fueled aircraft would have the same issues as hydrogen fueled rockets: they have to be very large to carry an economically useful amount of fuel.
There are other methods of storing hydrogen (developed for automobile prototypes); as I recall, the highest density of storage (kilograms of usable hydrogen per volume) has been with metal hydrides; these require neither very high or very low temperatures, and operate at reasonable pressures (rather than hundreds of atmospheres as common pressure bottles do) -- but they barely exceed the density of liquid hydrogen, and there's a penalty in the weight of an &quot;empty&quot; tank, which is still filled with a solid material when no more hydrogen can be drawn from it.
Beyond qustions of range, hydrogen is difficult to work with in terms of materials (it causes most metals to become brittle just from being in contact with hydrogen), it leaks through almost everything, and is so flammable that these unavoidable leaks are very prone to become invisible conflagrations (because a hydrogen fire contains nothing that glows from heat, the flame is practically invisible -- the bright flames of the Hindenburg were due to the envelope and its coating, not the hydrogen).  This is obviously irrelevant to engine efficiency, but very relevant to whether such an engine can last long enough in service to be economical to operate.