Similarly, gather as much evidence to determine why, based on the merits of your case, you were wrongfully denied your variation. This could be evidence that was both available to the police at the time of refusal and evidence which you have collated since.
This is important.
In defending an appeal the Police will, amongst other things they may cite, say that they based their decision on the information YOU provided to them in support of the variation. Their argument will be that they do not have a crystal ball and hence have no idea if you have other relevant information or factors that might help them determine the application. If you think they have ignored information you've sent them this is where you can demonstrate they had it in the first place.
As far as submitting any ADDITIONAL information this can be a little difficult as they will argue, & the courts are likely to agree, that if it was relevant you should have submitted it in the first place.
HOWEVER, you have two lines of response to this.
Firstly, that had they asked for additional information or given you an indication that they required more information before they were able to determine you application then you would, of course, have provided it.
Secondly, if it is information that addresses any of the points made, issues raised or reasons given in their refusal.
There is a third line you could also follow & this is any case law or other precedent you can find that supports your application & can be used against their decision.
The key, as already mentioned above, is to get advice from a qualified legal professional who specialises in firearms law.
Good luck.