Good, a clear and explicit answer. The relay's momentary clicking upon insertion is likely immaterial.
I'd diagnose as a open circuit in the relay control signal, either in the wiring, internal to the PCM , or a poor/bad connection at either end of the signal line.
Ignition off.
Disconnect connector C175B from the PCM.
Inspect pin 12 on both connectors for any evidence of corrosion, bent or retracted male pin, stretched female pin or anything else out of the ordinary.
If visual reveals nothing of consequence, perform a resistance check from harness connector pin 12 to AC Clutch Relay socket pin 1 (you labeled it as a blue dot in your photo). Should read zero ohms, manual says less than 5. Give the harness at both ends a good flex and wiggle test to check for any intermittent connections.
If nothing notable is found, it's likely an internal PCM failure.
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