No power for trailer brakes or running lights

You have resistance in the circuit. Most likely it is at the connector itself as there is water intrusion causing corrosion. However, the way to find this is to measure resistance from the junction block to the connector. You should have less then 1 ohm of resistance. I suspect you have slightly higher then this which is where you voltage is going.

I attached the wiring diagram so you can trace the wire from end to end.

Here is a guide that explains how to test wiring. Please let me know if you have questions.

https://www.2carpros.com/articles/how-to-check-wiring

Lastly if it is the connector, you can just replace the connector itself. Please reach out and let me know what you find. Thanks