Both Intel and Xilinx sell FPGAs with hard ARM cores inside so you can run real Linux while being able to interface with custom logic. Additionally, it's pretty common to create ARM, RISC-V, or PowerPC soft cores in the FPGA when there is no hard cores available. These mimic the real cores and will run software while allowing for things like custom instructions that can take advantage of the flexibility of FPGA fabric. The Xilinx Zynq and Intel Cyclone V have options for hard ARM cores. There are various designs of boards out there you can buy that implement Arduino or Raspberry Pi shield compatibility. The XUP PYNQ-Z2 supports both interfaces and runs a Zynq-7000 with a real ARM core.

You can do other things with soft cores that are not possible with an off the shelf CPU like triple mode redundancy. This is when you run a lot of the logic in triplicate and vote on the results to prevent a bit flip from messing up the software. This is common for space-based CPUs that are running on FPGAs. It's expensive to design a new chip in a very small run so it's much cheaper to just put the core on an off the shelf FPGA and use the rest of the FPGA fabric for custom logic functions.