It's very strange, yesterday when I did this, I tried to power up the Raspberry once but still the same problem. I think that must be the same as restart the GPSD. Today when I spend some time on this problem, I start the system and now the time is correct.
That's very good, but I must trust the system, out in the cold dark with only a pad I can't do very much. From my friend I got the commando I need to run to restart the GPSD: sudo systemctl restart gpsd. I write it down so I can do this out on the field.