Can turboprops use a recuperator?
An aircraft engine is a thermodynamic device, that converts heat to mechanical power. A 'virtual' air parcel entering the engine goes through a thermodynamic cycle, which determines the bulk of the power plant efficiency, which include quite a big chuck of irreversibele air friction, with a minor role for bearing losses.
The engineering optimisation process however includes a lot more than just power plant efficiency. At the bottom line, the optimisation is tailored towards generating the most dollars of profit per unit of time, but a more optimistic view is that the engineer will optimise the aircraft fuel efficiency as a whole. In practice, this results in finding a balance between power-to-weight ratio and fuel efficiency, as additional weight increases the power required.
A recuperator is a heat exchanger between the compressor discharge air and the engine exhaust air*. Its goal is to replace part of the energy required from combustion of fuel by energy extracted from the exhaust, and does not modify the thermodynamic cycle in any way. I looked up some (approximated) data from a PT-6 turboprop at 850shp: [source, copied from training manual]

Ambient: 300K
Compressor discharge: 600K
Turbine inlet: 1200K
Exhaust: 800K

A recuperator could in theory cool the exhaust to 600K and heat the compressed air to 800K (ideal infinitely long counterflow heat exchanger) for a delta of 200K. The total desired temperature increase from compressor discharge to turbine inlet is 600K (1200K-600K), so we can reduce the fuel flow required by 1/3rd (only 400K delta still required). That seems excellent!
In practice, heat exchangers are bulky (air is a terrible fluid to transport heat). Furthermore, it's only operating on a 200K difference. The real question then is whether a real-world air-to-air heat exchanger (likely of cross-flow type, like an automotive intercooler, so less efficient than a counter flow heat exchanger) can be made so efficient that it does not introduce more losses than gains. Historic evidence suggest this is unlikely, especially if cost is considered (the material must be able to withstand 800K).
Furthermore, if we're going down the road of heat exchangers, different options are available. For example, an intercooler (cooling between compressor stages, which actually improves the thermodynamic cycle) could work on a higher temperature differential as well as lower absolute temperature, and would only present a single flow obstruction when the air is still nicely compact (power required is volume flow times pressure drop!) inside the engine. However, historic evidence again suggests that this is not cost-efficient, and instead seems to favour increasing the combustion chamber temperature and pressure.
*You may wonder, why not heat the intake air with exhaust air? We would work on a much larger temperature difference! In fact, this would be counterproductive, as the thermodynamic cycle operates on the difference between high and low temperature. Increasing the temperature at the inlet disproportionately increases the power required to compress the air, and does not result in an increase in turbine power (which in the end operates on a pressure differential, and cannot extract work from hot air at ambient pressure). There's a reason we use intercoolers, not inter-heaters in compressors.