Can I use the second power of either spells after I've already displayed them for their initial power? That is, suppose I display Sphere of fire and use the fire power to roll my Arcane skill +1d6 on a combat check. On a combat check later in the turn, can I then discard it to roll Arcane skill + 2d6?
Yes. You can also repeatedly use the first power. I often display sphere of fire and use it repeatedly during my turn, then when I know I'm making my last combat check for the turn, I discard it for the extra oomph.
Yeah, that's exactly what the card says. For some reason I got in in my head that I could only get the additional dice if I discarded the card when I played it from my hand.
Just to add, the card reads
Display this card. While this card is displayed, for your combat check you may use your arcane or divine skill+1d6; you may additionally discard this card to add another 2d6. At the end of your turn, if you do not have either the Arcane or Divine skill, banish this card; otherwise, attempt an Arcane or Divine 11 check. If you succeed, recharge this card; if you fail, discard it.
The important thing here is that the card tells you first to display it, and then "While this card is displayed" it has the rest of the text which gives you the choice of using arcane/divine+1d6 and also the option to discard it for the extra 2d6. If they didn't want you to be able to do that, then it would've phrased it differently so that you were either displaying it or discarding it for the ability. The way it is currently written you always display it, then you continue to read the rest of the card text.