  We investigate how the motive of hyper-K\"ahler varieties is controlled by
weight-2 (or surface-like) motives via tensor operations. In the first part, we
study the Voevodsky motive of singular moduli spaces of semistable sheaves on
K3 and abelian surfaces as well as the Chow motive of their crepant
resolutions, when they exist. We show that these motives are in the tensor
subcategory generated by the motive of the surface, provided that a crepant
resolution exists. This extends a recent result of B\"ulles to the O'Grady-10
situation. In the non-commutative setting, similar results are proved for the
Chow motive of moduli spaces of stable objects of the K3 category of a cubic
fourfold. As a consequence, we provide abundant examples of hyper-K\"ahler
varieties of O'Grady-10 deformation type satisfying the standard conjectures.
In the second part, we study the Andr\'{e} motive of projective hyper-K\"ahler
varieties. We attach to any such variety its defect group, an algebraic group
which acts on the cohomology and measures the difference between the full
motive and its weight-2 part. When the second Betti number is not 3, we show
that the defect group is a natural complement of the Mumford--Tate group inside
the motivic Galois group, and that it is deformation invariant. We prove the
triviality of this group for all known examples of projective hyper-K\"ahler
varieties, so that in each case the full motive is controlled by its weight-2
part. As applications, we show that for any variety motivated by a product of
known hyper-K\"ahler varieties, all Hodge and Tate classes are motivated, the
motivated Mumford--Tate conjecture holds, and the Andr\'e motive is abelian.
This last point completes a recent work of Soldatenkov and provides a different
proof for some of his results.
