  We consider the problem of determining the asymptotic order of the Gelfand
numbers of mixed-(quasi-)norm embeddings $\ell^b_p(\ell^d_q) \hookrightarrow
\ell^b_r(\ell^d_u)$ given that $p \leq r$ and $q \leq u$, with emphasis on
cases with $p\leq 1$ and/or $q\leq 1$. These cases turn out to be related to
structured sparsity. We obtain sharp bounds in a number of interesting
parameter constellations. Our new matching bounds for the Gelfand numbers of
the embeddings of $\ell_1^b(\ell_2^d)$ and $\ell_2^b(\ell_1^d)$ into
$\ell_2^b(\ell_2^d)$ imply optimality assertions for the recovery of
block-sparse and sparse-in-levels vectors, respectively. In addition, we apply
the sharp estimates for $\ell^b_p(\ell^d_q)$-spaces to obtain new two-sided
estimates for the Gelfand numbers of multivariate Besov space embeddings in
regimes of small mixed smoothness. It turns out that in some particular cases
these estimates show the same asymptotic behaviour as in the univariate
situation. In the remaining cases they differ at most by a $\log\log$ factor
from the univariate bound.
