How does the input impedance of an amplifier effect it sonically?

That depends on what is meant by 'struggling' as there can be several issues. One is that the amp needs a lot of input voltage and this has nothing to do with the input impedance. The other is that if the preamp has an output coupling capacitor, its output impedance at 20Hz can be quite a lot higher than it is at 1000Hz, and this can affect the bass if that 1/10th rule I mentioned is not observed- while taking the 20Hz output impedance into account.

So I assume 1/10 isn’t going to enough of a safety net when matching preamp to amplifier input impedance. Since impedance can vary with frequency and most manufacturers don’t provide accurate numbers 1/20 sounds like a safer bet when matching pre to amp.

@hiendmmoe  If the manufacturer can't provide that 20Hz output impedance value, you probably should look elsewhere! The 20Hz value can be a multiple of the 1KHz value; it could easily be 10x the value at 20Hz. We got around this issue with our preamps by direct-coupling and so the output impedance of our balanced preamps is the same at all frequencies. 

Is it then more important to know the correct value of the output impedance of the pre-amp at 20HZ than to know how accurate the input impedance of the amp is.
As for the amplifier input impedance is 20HZ also the number I should look for when a specific input impedance is given. If I understand correctly, 20hz is where the amplifier will show its  minimum value and any higher specification (frequency) given for impedance  is not its true value.

As for the amplifier input impedance is 20HZ also the number I should look for when a specific input impedance is given. If I understand correctly, 20hz is where the amplifier will show its  minimum value and any higher specification (frequency) given for impedance  is not its true value.

That depends on the amplifier. If a direct-coupled input, the impedance will be the same at all frequencies. If capacitively coupled, its input impedance at 20Hz may be a bit higher, although any designer worth his salt would have set the timing constants of the input so low in this case that the 20Hz impedance would be nearly the same as that at 1KHz.