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My detector has issues with multiple measurements in succession, i.e. some of the strain is retained between measurements.

The Arduino Uno I'm using has this reset button on the board, allowing me to rerun the program and tare the device again.

I don't think there is anything in the rules against that, but just for confirmation, can I press that button to restart the program (and tare the device) before measuring each time?

And I do have enough time to restart the program and measure the mass in one minute, so no worries about that.

You have one minute to measure each sample. Restarting code, taking a zero sample etc. all seem to be reasonable things you could chose to do in that minute.