The circuit seems to be of a flicker free ballast with possibly 1300x series transistor. But in these fluorescent lamp circuit, these ferrite cores are inductors. Some call it a "choke inductor". It might have 4 or more pins below, but there's only one wire wound all over the core. Rest of the pins are either left unused, or might be fused with its neighbor pin. There's no secondary wiring in it.
But you can use the ferrite core to design your own trafo, depending on its size, requirement and operating frequency.
Note: Ballasts are not power adapters, like the one in laptop. They just convert low frequency input (50Hz) to high frequency output (30kHz to 50kHz) for efficiency, and is specially designed for fluorescent lamps. ;-)