Bob -
Please DO NOT apply 5 volts to the device if you have the battery connected directly to the power switch!!
You WILL end up with a battery fire.
The only reason the unit is working is because there is a low drop out regulator that is converting that 3.7 volts down to 3.3 volts.
HOWEVER - your SA612 mixers are no longer getting the 5 volts they need to operate with and your readings will not be accurate.
The 5v at the USB socket drives the charger/inverter chip that simultaneously charges the battery and provides 5 volts to the power switch to run the Nano.
I've used an existing forum photo and added the spot (red) where you can manually connect +5V to R40 at Pin 1 of the IP5303 chip.
But first, put the battery back where is is supposed to be connected.
you can use a barrel jack and a 5V wall adapter and you're good to go.