Pulsed propulsion is certainly possible, but there are a bunch of engineering issues that mean such a design would never be more efficient than a comparable constant-flow chemical rocket. For nuclear propulsion, the engineering challenges are somewhat different.
There are numerous problems with pulsed chemical propulsion.
A constant flow engine is providing a (more or less) constant amount of acceleration. Engineering the rocket to sustain the stresses of, say, 3-4 Gs of sustained acceleration is much easier than finding a design that will survive 20 Gs of intermittent acceleration. You can put dampers and shock absorbers onto the end of the rocket, but that is additional weight. All the components that receive the full brunt of the shock must be strengthened accordingly, and will therefore also contribute extra mass to the craft.
As you know, rockets go forward by producing high-velocity combustion products and channeling them all to go backwards. A constant flow engine is very good at doing this - from the combustion chamber to the throat to the nozzle, the flow becomes supersonic and is channeled in almost completely the same direction.
With pulsed projection, the explosion has to happen outside of the chamber. Only the fraction of the explosion that hits the shock plate contributes to the forward thrust of the craft. The rest gets lost sideways and backwards. This problem could be mitigated somewhat by using appropriately shaped charges, but you will still get quite a lot of inefficiency.
3. Propellant choice
This one I consider relatively minor, but relying on a supersonic detonation rather than deflagration limits the choice of propellant. This may be surprising, but explosives like HMX or TNT have a much lower energy capacity than kerosene/LOX, and therefore a lower max Specific Impulse. Kerosene/LOX just happens to combust slower than those explosives and cannot create a supersonic shockwave. I don't know for sure if a supersonic shockwave is required for pulsed propulsion or whether such a system is feasible for propulsion in vaccuum.
1950's slomo tests of pulsed propulsion using conventional explosives
So why do designs exist for pulsed nuclear propulsion? After all, any design would still suffer from the same problems as with chemical propulsion.
The answer is that sustaining a constant-flow nuclear reaction is much harder than repeatedly detonating nuclear bombs. Nukes benefit greatly from economies of scale, and so a few larger bombs would be much more efficient than the same mass worth of smaller bombs. Proposals exist for such constant-flow nuclear propulsion involving enriched uranium salts dissolved in water (See Zubrin's Salt water rocket), but the engineering difficulties involved would be immense. Sustaining a nuclear reaction (which is inherently unstable, easily fizzing out or running away with even small variations in neutron flux) in a medium involving turbulent flows of water, would be...difficult.
A nuclear pulsed-propulsion vehicle would be comparatively much easier to design and build than a nuclear salt water rocket, yet would suffer from the same inefficiencies as chemical pulsed propulsion. Theoretical calculations for Salt Water rockets generally give Isp's much higher than those for Project Orion-style spacecraft.