The lighter should be a hot connection...works even when the key is turned
off.

The windshield washer would only work when the key is on. In other words,
they shouldn't be on the same circuit.

The door to your fuses should tell you which one is for the windshield
washer. Pull that one and check it for continuity. It might be the pump and
not a fuse.

There is only one wire running to the cigarette lighter, remove your ash
tray and you should be able to see if it's still connected to the female end
of the outlet. Check to see if you have 12 volts there.

Did the two items fail at the same time?  Actually, both of these are fairly
common failures in older vehicles.  I'd venture to say that the two are not
related.  Most likely, you'll need to buy a washer pump.  On the cigarette
lighter, have you checked to make sure the wiring is intact on the back of
the socket?  If it's used a lot, the connections tend to get corroded due to
heat.

CJB