What you're writing appears to me to be "science fiction".
There are at least two kinds.
Hard science fiction:
Hard science fiction is a category of science fiction characterized by an emphasis on scientific accuracy.
Soft science fiction:
Soft science fiction, or soft SF, is a category of science fiction with two different definitions.
- It explores the "soft" sciences, and especially the social sciences (for example, anthropology, sociology, or psychology), rather than engineering or the "hard" sciences (for example, physics, astronomy, or chemistry).
- It is not scientifically accurate or plausible; the opposite of hard science fiction.
Soft science fiction of either type is often more concerned with character and speculative societies, rather than speculative science or engineering. The term first appeared in the late 1970s and is attributed to Australian literary scholar Peter Nicholls.
There are good examples of both kinds in the world.
As a reader of SF I've read and enjoyed both kinds (and Fantasy too).
I tend to have favourite authors (as an avid reader if I find a book I like I might read everything by that author).
But does my story need to be scientifically accurate or plausible?
I don't think accuracy is as as important as with "historical fiction" -- I mean of course the world is imagined, even when it makes an effort to be hard-science (perhaps as a way to make it more interesting to people who are interested in science).
Even supposedly-hard science fiction requires some "handwavium" a story which introduces space elevators to earth orbit, or linear accelerators on the moon, introduces a concept but ignores real-world practical difficulties. It's fiction.
If there's not much science in the book then I'd hope there's something else.
Will I lose readers because all they can think of is "that would never happen"? Can a story like mine with inaccurate/nonexistent science still be appealing?
For example one book I read once was like the one you suggested: i.e. a world with mutations. Society was rural/agricultural, no longer mostly-urban, and any mutants were killed. So anyway, there, some children were travelling alone somewhere with a "great horse" -- a huge horse, a cart horse, I forget whether it was mutated or just bred, a big breed, but its being so unusually big meant that the people in the lands they were passing through would kill it if they found it, so they were trying to stay hidden. And I wanted to know more: about the children, the horse, the land they were travelling through, what would happen, and so on.
Edit to add: it turns out that I was remembering The Chrysalids. The post-industrial characters don't really know what causes the mutations:
The inhabitants of post-apocalypse Labrador have vague knowledge of the "Old People", a technologically advanced civilisation they believe was destroyed when God sent "Tribulation" to the world to punish their forebears' sins. The inhabitants practise a form of fundamentalist Christianity; they believe that to follow God's word and prevent another Tribulation, they must preserve absolute normality among the surviving humans, plants and animals, and therefore practice eugenics.
The scientific cause of the tribulation is hinted at for the reader's benefit ...
Though the nature of "Tribulation" is not explicitly stated, it is implied ...
... but knowing (being told, or seeing from the characters' perspective) that it was "God-sent" makes me dial-back the scepticism and get on with reading the story on its own terms. I guess I was sophisticated reader at the age when I read that, but when the author effectively says, "this is a Deus ex machina") then as a reader I know to treat that as background scenery -- like people watching a play know that they're meant to not see the Kuroko.
It wasn't even a plot-point, it was a fait accompli, past history, no need nor opportunity to explore that premise any further.