This looks like a common problem when relays and electronics are both included in the same system. The cause will be one of two things. With a decent oscilloscope, you can get to a firm diagnosis. Either one or both of these could be true.
1) There is a signal spike being induced on the signal wires to the speedo.
Or
2) The relay coil operation is changing the supply voltage to the speedo.
Without knowing more about the signal shape to the speedo, I would be reluctant to mess about with it although I have seen reference to people fitting a small ferrite bead here (sorry, cannot remember where I saw this).
A safe experiment, is to wire a small capacitor across the power supply to the speedometer - unless you short things out, you cannot make things worse. As to value, I would start with 0.1 µF wired between speedometer power in and speedometer earth as close as practicable to the device. If this makes any improvement at all, gradually increase the value and see what happens. If you get to 0.5 µF and there has been no improvement, then this isn’t the problem. Make sure you buy capacitors with a voltage rating of at least 35V as the spike might be a fair size. Buying 5 0.1 µF will work as an experiment - when you wire them in parallel, the values add so 2 x 0.1 µF wired in parallel gives 0.2 µF.
