The official setting for Freedom City from 2E on, depicted in the illustrations, is Steve Kenson's "Freedom City" setting. While there are various forms of mutation listed in the section on origins, most of them are engineered. The closest parallel to something like Marvel mutants are the mutations caused by the Terminus energies from the failed invasion, suggested as one of the ways to introduce a common origin for powers in this world.
The unleashed energies of the Terminus might have led
to numerous origins and new superhumans in the world.
Mutations could be on the rise, with people understandably
worried about these “Terminus mutants” and their
potential links with the invaders.
This is a standard origin for a few of the play-by-post settings, Terminus Babies.
Officially recorded as Terminus Energy Mutation Syndrome (TEMS) by AEGIS, ASTRO Labs and the Freedom League, the empowerment of humanity by the energies of the Terminus is more commonly known as Terminus Babies, or T-Babies. It is most common in those exposed to low levels of Terminus Energies while in utero, though higher levels of exposure have been known to have effect upon children and even adults. The precise mechanism of the mutation and subsequent development of supernatural abilities is not well understood though extensive testing has revealed that there is no lingering connection to the Terminus only the mark of its entropic energies upon the cells of the afflicted. Most affected develop powers almost immediatly [sic] within a weeks time for most older individuals and at birth for those exposed in utero.
However, this is all relatively speculative material. For what it's worth, the section on "Mutations" as an origin focuses on Designer Drugs (canonical versions of which give superpowers for a short amount of time, but occasionally result in a permanent effect with certain body chemistries), the DNAscent Process engineered by Labyrinth, and Manaka Root (basically, the "heart-shaped herb" from Black Panther).