As you can see in the statistics in the images below, the input impedance is steady at around 10Mohm both at 1.9V and at 2.3V. Thus, the relay does not seem to affect input impedance.
Thanks Amaruk. I too have revisited the measurement.
Still using the voltage across a 1Mohm resistor to calculate current flow into the Siglent when measuring 1.8V and 3.6V. i.e. either side of the relay "click".
What is different this time is that I am using a different millivoltmeter. I measured the input resistance of this millivoltmeter with the Siglent to be a suprisingly low 800Kohm which obviously loads the 1Mohm current sensing resistor quite a bit.
Because I have learned never to trust my math

I created an LTSpice model of all of this and the results - both before the relay click and after the relay click, the LTSpice numbers all add up to 10Mohm input of the Siglent, matching the mV readings taken on the meter.
So no impedance change on the Siglent when the relay clicks.
I don't know what was going on with the VC8145 but on mV it was giving some very strange results. I did use a short twisted measuring probe set but maybe it was still some noise getting in.
If my latest measurement was the only one, I would probably accept it with nagging doubts, but it ties in perfectly with results obtained by Amaruk which seem beyond doubt anyway.
So definitely no funny impedance change problem.
When that relay clicks, my VC8145 does give some totally different indication of impedance so whether it is a bug or not, it is definitely altering something. Based on our latest measurements I am at a loss to understand exactly what.
The fact remains that on a manual range or auto range, if I want to measure a voltage between 2.2V and 6 volts , I have to put up with a relay click, and a delay every time I apply the probe to the DUT. Doesn't make me proud to own, nor happy to use, this meter.
Regards
Eloso