I use an Al. pole that is a bit shepherds crook shaped.
I made up a long length of copper pipe, bent into a shepherds crook
with the end hammered flat to make a bit of a jet. It works, but
chiefly it was a source of great entertainment for the neighbours,
who all came out to watch me washing the mud (mostly sand) out of
the gutters, which went all down the front of the house and all
over me. It all hosed off, but they still found it very amusing.
Unless you want to be washing the sand out of your eyes for the
next week, goggles (preferably sealed swimming goggles) are essential.
Don't use high pressure - you may find it blows the gutter seals
out if they're in less than perfect condition.
If you can, disconnect downpipe from soakaway while you do this
so you don't wash the muck into it.
The shepherds crook pipe is about 4.5m long 15mm copper, and this
is too long a length of copper pipe to be self-supporting, so I
had to hold it dead vertical to move it, particularly when full
of water. Something stiffer would have been better.
However, now that I have slowly completed replacement of all the
guttering, nothing gets stuck in the replaced gutters, so I don't
have to do it anymore. Don't know if that's because the guttering
has a more polished finish, or if it's because I got the fall on
it correct. There isn't a lot of moss on my roof, and when I was
up there on scaffolding, I did knock off the few large clumps
that were there, although some new ones have formed since.