Which is more efficient: a jet engine or a rocket-powered engine?
Jet engines are far more efficient. The metric to compare them is 'Thrust-specific fuel consumption', i.e. the amount of fuel used to produce one unit of thrust. Fuel consumption per distance traveled is proportional to this. 

A few figures from that page:  

Rocket engine: Space Shuttle main engine, 225 g/kN.s

Jet engine in afterburner: EJ-200: 48 g/kN.s

Jet engine, dry: CF-6: 17 g/kN.s

So a rocket uses ~10 times as much fuel to produce the same thrust. Even if you add afterburning, a jet is more efficient. 

During and after World War 2, there was some interest in rocket-engined aircraft. The Me-163 saw service in the Luftwaffe. 

The interest continued for a decade or so, until jet engines with afterburners were developed. One exponent of that interest was the Saunders-Roe SR-53, a short-range interceptor with both a jet engine and a rocket. 

It was developed in response to an RAF requirement for an aircraft that could take off and get to high altitudes very quickly; more quickly than could be attained with the jet engines of the day.


Jet: Armstrong Siddeley Viper, 1640 lb thrust. 
Rocket: De Havilland Spectre, 8000 lb thrust. 

Fuel carried:
jet: 500 lbs
rocket: 10,500 lbs

Endurance:
jet: 45 minutes
rocket: up to 7 minutes  

speed:
jet: 180 kts
rocket: accelerate to Mach 2 once, then the fuel ran out

(source: mostly 'British experimental jet aircraft', Barrie Hygate.)

It was superseded by the EE. Lightning, which had 2 afterburning jets. It had 4 times the range despite carrying less fuel and more payload.