  The Ablowitz-Ladik (AL) equations are distinguished integrable
discretizations of the focusing and defocusing nonlinear Schr\"odinger (NLS)
equations. In a previous paper (arXiv:2305.04857) we have studied the effect of
the modulation instability of the homogeneous background solution of the AL
equations in the periodic setting, showing in particular that both models
exhibit instability properties, and studying, in terms of elementary functions,
how a generic periodic perturbation of the unstable background evolves into a
recurrence of anomalous waves (AWs). Using the finite gap method, in this paper
we extend the recently developed perturbation theory for periodic NLS AWs to
lattice equations, studying the effect of physically relevant perturbations of
the $AL$ equations on the AW recurrence, like: linear loss, gain, and/or
Hamiltonian corrections, in the simplest case of one unstable mode. We show
that these small perturbations induce $O(1)$ effects on the periodic AW
dynamics, generating three distinguished asymptotic patterns. Since dissipation
and higher order Hamiltonian corrections can hardly be avoided in natural
phenomena involving AWs, and since these perturbations induce $O(1)$ effects on
the periodic AW dynamics, we expect that the asymptotic states described
analytically in this paper will play a basic role in the theory of periodic AWs
in natural phenomena described by discrete systems. The quantitative agreement
between the analytic formulas of this paper and numerical experiments is
excellent.
