  We study the deformation theory of rational curves on primitive symplectic
varieties and show that if the rational curves cover a divisor, then, as in the
smooth case, they deform along their Hodge locus in the universal locally
trivial deformation. As applications, we extend Markman's deformation
invariance of prime exceptional divisors along their Hodge locus to this
singular framework and provide existence results for uniruled ample divisors on
primitive symplectic varieties which are locally trivial deformations of any
moduli space of semistable objects on a projective $K3$ or fibers of the
Albanese map of those on an abelian surface. We also present an application to
the existence of prime exceptional divisors.
