What kind of aperture are you working around in the lighting available? There are two things that can happen. The first thing is that the change in brightness is because the lens actually IS closing the iris slightly the you zoom in, which sometimes happens when the exposure is on auto, because the iris is being set by the average content of the image – which is with ice, lots of white – as you zoom in filling the frame with non-white things, this means the picture does on average appear darker, so the usual thing is the camera counteracts this by opening the iris, but if the iris is already near wide open, it can’t do it. Occasional zooms are good – but your single camera means the dreaded trombone zoom – often wide max in to narrow max, then back again and that’s difficult to watch for a whole game.
However, the more likely problem is simply the optics in the lens – it’s not a sophisticated lens and is an optical compromise and as the focal length increases, the light transmission drops. In fact, as the manufacturers try to offer more zoom range, it gets worse! If there is plenty of light, the auto exposure system can often reduce the impact, but with loads of white, it’s possible it’s making it even worse. try manual exposure and test – it will either improve or get horribly worse, and I think on your camera it isn’t easy to control.
In practice, sorting it in the edit might just be easier.
For what it’s worth, the constant zooming required when working single camera gets very distracting, especially when not always starts with a lurch. A cheap solution is to use a wide angle camera – one of those cheap Chinese Go-pro-esque ones will do, attached to the head somehow that tracks with you and then you concentrate on a closer shot all the time, limiting your zooming requirements, and cutting to the wide shot while you zoom in – so you forget the slow gradual zooms and get in there quickly and do the cuts in the edit. The other tricks with zooming are perhaps impossible on your camera as the zoom is a little basic – but it’s common in broadcast to track the zoom at the same rate the person in the frame is coming forward or back – so the player at the other end coming towards the camera fills the frame, and you zoom out as he gets closer, maintaining his height in the frame – I don’t think the speed on your zoom will allow this, and it needs to be done pretty accurately.
.
