Why does a turboprop lag?
Turboprop engines typically have a core which includes a compressor and turbine that rotate as a unit (although an engine might have more than one such "spool"). The gas generated by the core drives a "power" turbine which, via a gearbox, drives the propeller, which by the way has a non-trivial rotating mass. The core and power sections rotate independently of each other. So, upon a commanded power increase, the core will first spin up, throwing more gas at the power turbine which will then speed up. The delay is therefore due to the rotating mass of the propeller and the fact that it rotates independently of the engine core.

For what it's worth: turboprop engines are just a specific application of a turboshaft engine - where the energy of the gas produced by the core is captured through a power turbine to drive a shaft, which could be used to power anything - propeller, helicopter rotor, electric generator, etc.

I should add that there are exceptions to the above, like the Garrett TPE331 which have only a single shaft.