What is Besauce's core skill?

Charged Geyser costs zero EP, so you can use it when your bar is close to empty. That matters most below 20 EP, where the free cast is confirmed. A support that can still act when the team has nothing left is the whole point of bringing Besauce.

What the skill gives back is energy. Charged Geyser puts an EP regeneration buff on the whole team for 12 seconds, restoring 1.5 EP per tick. In practice you cast it when your damage dealers run low, then let them spend the refilled EP on their own skills. Firing it at a full bar wastes most of what it does.

You will also see the name written as B-Sauce, and Charged Geyser is listed as Beeswax's core skill too. If a team guide or skill list uses one of those names, it is talking about the same support kit built around Charged Geyser.

Upgrading Charged Geyser

How a normal, non-Alpha Aniimo's Core Skill is upgraded is not settled. Three routes are described: spending 10 Marks of the Strong, forming a contract with a Legendary Held Item, or using Prismatic Stones together with capability fruit, which upgrade the Core Skill once to boost its damage and add utility. Check which of these your build of the game actually offers before you hoard materials for Besauce.

Not settled: The zero EP cost is only confirmed when you cast Charged Geyser below 20 EP. Whether it stays free above that is not settled.