Sorcerer-kings creating an army of "half-golem" to work (or fight) as slaves. Their creation would be with the soul of punished criminals, or a special reward for the most loyal agents. Some creations would be voluntary, but with the promise of a reward reincarnating into a better biological body.
My suggestion for constructs PCs in Dark Sun is adding something like the disquiet from Prometheus: the Created (Chronicles of Darkness).
Disquiet is the name given by Prometheans to the invisible aura they emit which causes humans, animals and even the land itself to reject them. To a Promethean, it feels as if the whole world has renounced them. Disquiet has a deleterious effect on the natural world, and is the chief cause of a...
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Other idea is constructs need food and water for a "biological exosuit" used as costume in the urban zones. Or constructs created by the sorcerer-kings need some special alchemic oil. This flaw is intentional to avoid they could become too independient.
Or the constructs PCs need eating and drinking for pleasure, to keep their mental health.
* Lots of players love the idea of PC dragons, but there aren't monster classes in 5ed. The closest option would to get a dragonborn and a new class style dragon disciple, about acquiring draconic traits. Dracolyte?
Some time I thought about a new setting mixing "Council of Wyrms" and Cris Perkins' Iomandra homemade setting, but Io's blood islands would be a cluster of demiplanes within the elemental chaos of limbo. Each domain ruled by a dragon overlord, and this would allow space for all dragons from previous editions. For example the cobra dragons would be the rulers of a domain inhabitated by yuan-ti.
* Fandom has created homemade version of PC species based in famous franchises.
* Mutants are more as lineage than a true "specie".