Like a dummy, I fitted mine with a hose, with water pumped from the raw water outlet.... just because it's in the instructions.
My shaft log is 10 inches long and doesn't have a cutlass bearing (it's on a strut) so there's lots of water flow to cool the joint. I'll be removing all that at next haul out and fitting a plug.
The hose CANNOT be pulled to the side of the hull if installed, it must be centerline and at least 12 inches above the waterline. If you can't do that, plug it or use a different shaft seal. If installed with water injection, use the outlet side of the raw water vented loop.