  Nonlinearity and disorder are key players in vibrational lattice dynamics,
responsible for localization and delocalization phenomena. $q$-Breathers --
periodic orbits in nonlinear lattices, exponentially localized in the
reciprocal linear mode space -- is a fundamental class of nonlinear oscillatory
modes, currently found in disorder-free systems. In this paper we generalize
the concept of $q$-breathers to the case of weak disorder, taking the Discrete
Nonlinear Schr\"{o}dinger chain as an example. We show that $q$-breathers
retain exponential localization near the central mode, provided that disorder
is sufficiently small. We analyze statistical properties of the instability
threshold and uncover its sensitive dependence on a particular realization.
Remarkably, the threshold can be intentionally increased or decreased by
specifically arranged inhomogeneities. This effect allows us to formulate an
approach to controlling the energy flow between the modes. The relevance to
other model arrays and experiments with miniature mechanical lattices, light
and matter waves propagation in optical potentials is discussed.
