Does a higher gauge electrical wire impact voltage level?

OP : Yes, wire gauge can (in theory) impact voltage at the load.   A perfect wire has no voltage difference, and would have the same voltage at the load as at the source, whether we are talking AC power or speakers.

In AC power transmission in a home, 1-2 V per leg loss is normal.  Meaning, 2-4 V drop at the appliance relative to the panel.  A higher gauge wire than required can reduce this. 

As I mentioned before, the voltage loss (i.e. drop) is proportional to the current.  It is also proportional to the wire length.  For the same wiring, a high current device on the end of a long line will suffer more drop vs. a low current or short run.

For speaker cables it is extremely rare to expect voltage drops more than a couple of tenths of volts with anything equal to or better than 14 gauge cables.

Speaker cables? Got anything else extraneous to the OP?

Wire gauge will not affect no-load voltage.  Your reported voltage swings are very large and do not reflect well on your local utility's ability to provide you reliable source of electricity.

Voltage swings like this are likely accompanied by other imperfections such as noise. 

You are a prime candidate for a power regeneration (rectification/inversion) unit. 

@jasonbourne52  Erik's statement is correct.  Lower gauge wire has a higher number.  So if I have a number 12 wire and want to go to a higher gauge, I will go to a 10, not a 14.

@carlsbad

Excellent observations.

I would also question the accuracy of the test device. I have one that reads 125 while another reads 118, parallel OR in series.

In the end the voltage supply SHOULD have imperceptible effect on SQ so what is the concern?

@fuzztone 

Indeed, the voltage is not important to most ss amps.  however, the noise and other things that challenge your amp to produce quality sound also are likely increased with poor voltage supply.

many voltage meters can have a zero offset, but if he is using the same meter for all measurements, he observed spread of 13 volts is probably real.

Tube amps can benefit from constant voltage, some more than others.  My power regenerator provides 119.9 volts 24/7/365 so I never have to think about it.

Jerry