Most decent sized designs, hard or soft, have little processors embedded in them in addition to their core logic. It's an interesting space/time/ease of development trade-off between a special little processor, and doing the same work in just logic.

GPUs are pretty well documented (relatively speaking), so they make a good case study. Generally special processors will handle FIFO pulling (so the part that reads the command lists), DMA engines, power management, video codecs, DRM key management, and some other miscellaneous pieces (like run this code on the GPU' on certain interrupts, instead of interrupting the CPU). And that's all in addition to the shader cores you normally think of as 'the GPU'.

In the past, I've used simple processor cores in FPGAs for motor control.

I don't have good references handy where I am right now, but Hotchips a couple of years ago had many examples, including a Memcache accelerator.

A more down-to-earth example shipped with an Arrow FPGA dev kit I got: they took a software and ported MPG123 (mp3 decoding). They then profiled it and isolated a candidate for acceleration (some moderately wide integer operation). The result saved meaningful amount of CPU cycles and power. (The FPGA board was battery driven, which is still unusual).