  We consider deformations of sequences of cluster mutations in finite type
cluster algebras, which destroy the Laurent property but preserve the
presymplectic structure defined by the exchange matrix. The simplest example is
the Lyness 5-cycle, arising from the cluster algebra of type $A_2$: this
deforms to the Lyness family of integrable symplectic maps in the plane. For
types $A_3$ and $A_4$ we find suitable conditions such that the deformation
produces a two-parameter family of Liouville integrable maps (in dimensions two
and four, respectively). We also perform Laurentification for these maps, by
lifting them to a higher-dimensional space of tau functions with a cluster
algebra structure, where the Laurent property is restored. More general types
of deformed mutations associated with affine Dynkin quivers are shown to
correspond to four-dimensional symplectic maps arising as reductions of the
discrete sine-Gordon equation.
