Okay... I'll have a go, I guess...
The magic formula for Mach is:
M = { (2/(g -1)) [ ( 1 + q c/p )(g -1 )/g- 1 ] } 0.5
Therefore, Mach is only a function of measured pressures (static pressure p and impact pressure qc) and.... gamma. Which is a constant 1.4 for air, if you assume that the air through which the aircraft is moving is calorically perfect (another assumption of the standard atmosphere). The above equation is actually derived from the compressible Bernoulli equation, which makes the assumption of isentropic flow, i.e. no entropy change or heat addition.
But, in reality, do those assumptions hold true? In reality, there would be heat transfer to/from the air flow, air is not really calorically perfect (but close enough) and the value of gamma is probably not constant for extremes of temperature, etc. etc.
So temperature does probably have an effect on the Mach reading, but I'm not sure, based on the above considerations, how big of an effect it is.