Cherokee 140 Loss of Power in Precipitation
 

Any ideas what could cause this, quite a large drop in power while flying through precipitation?

Wet electrons/ leak path into the ignition wiring. Does it rapidly get all better when the out of the rain? There's also more moisture, allowing for carburetor icing, which I presume is a possibility though carb heat can also decrease engine power, but if there is ice most of the power will return when it is melted.

I would check that the induction air filter is clean and in good condition.

I would expect ignition issues to show up as intermittent firing, rough running or a drop similar to the magneto check (could be wrong of course). Carb icing would be one guess, followed by the intake air path as mentioned. I have had a similar event on a C150.

Assuming this is in the UK, I'd suspect carburettor icing first. Was carb heat selected and if so, did the engine then run rougher?
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Some airplanes, notably the C 182, will trap water on the front of the air filter in heavy rain. This reduces air flow into the engine and thus power. Selecting carb heat will resolve the issue. In any case trouble shooting any reduction in power in a carbureted engine should always start with selecting carb heat full on.

Yes, when selecting 'hot air' in the PA28-140 you also shut off the ram air intake and draw air from within the cowling bypassing the blocked ram air filter. Should you need to maintain hot air then the fuel/air mixture is enriched therefore, for extended periods, consider leaning the mixture to avoid fouling the spark plugs.