The trick is to understand what the mini computer in the camera wants to do. The exposure system wants to make the scene look like a nice medium gray. Under normal lighting, this works well. When the small subject is a big difference in brightness compared to the large background, the system starts to fall apart.
When the small subject is bright and the large background is dark, the camera wants to make everything brighter due to the dark background taking up so much space. This results in the small bright subject getting blown out in brightness.
When the small subject is dark and the large background is bright, the camera wants to make everything darker due to the bright background taking up so much space. This results in the small dark subject getting even darker.
The cool thing about the phone's camera is that it will show you the results on the screen before you take the picture. In manual mode, set the shutter speed to 1/500, which will initially be a really dark photo. Then keep scrolling up the ISO until the brightness looks right. Tapping on the screen to set the focus point will also help with the exposure as you are now telling the camera what the important part of the image is.
When the spotlight comes on, drop the ISO down. The catch with the spotlight on the ice is that when you get the subject to look good, the ice may still be too bright; but that is OK.