Over decades, various engine manufacturers have employed water + methanol (or other) injection to boost engine thrust, typically at take-off (or landing, if conveyed by Harrier).

Does flying through rain similarly yield an increase in engine thrust?

Turbine engines are limited by turbine temperature. The water injection lowers the temperature for a given fuel flow, and allow the fuel flow to be increased without exceeding the temperature limit. The increased thrust is due to burning more fuel. Flying thru rain wouldn't have this effect because there wouldn't be any increase in fuel delivered. Theoretically, there might be an increase resulting form the greater mass accelerated thru the engine , but I'm skeptical that this would be significant.

I think its not so much rain but more to do with humidity - an increase in mass
airflow. But then I'm talking JT3Ds and early 9Ds. What the modern donks do I
wouldn't know.

Thrust you want thrust?

Then push the throttle up and shed the water out the centrifugal separartor (i.e. the fan). The core doesn't get much water that way and the mass flow increase out the fan gives you all you want.

Now if you want nil thrust than pull back below flight idle and ram the water into the core compressor. The flame will cool off and the turbine will spool down and presto you're a glider

choices choices