Use a 1K resistor or something between 100 ohms and 5K and
place that across connector and measure V. Could be you have
a poor high resistance connection to fan connector. Or use 12V
lightbulb as load and measure V at connector. That would be better
as it will draw more current, better test of connections.
Is there any overload resettable fuse in series with fan, get a diagram
of electrical wiring for fan and relay circuit back to battery.
Use brake cleaner, better yet contact cleaner, on any connectors/
connections in wiring path to fan to clean them.
Regards, Dana.