Spell Mastery is "Accuracy for Mages", opposed to Spell Resistance instead of Dodge. Basically it allows your spells to land instead of getting resisted and its value is the 'to hit' of the spell minus SR.
Influence is a sort of resource which can be used in trade and for a few other reasons, but they aren't many. It pretty much represents your political clout. There are several ways to obtain it, although not all of them will be available at any time. Constructing non-city monuments (i.e. those resources you see on the map) nets you +1 Influence per turn. Completing a quest as Altar (or anyone with the Heroic faction trait) will net you +50 Influence per quest. Legacy of Serrene faction trait will grant you Influence per completed caravan. The Missionary faction-unique building, like monuments, gives +1 Influence per season. Then you also have the option to trade for it, just as any other resource. I may be forgetting other sources besides the occasional quest-reward (specific to the quest as an option, and stacks with Heroic).
