  This paper is a direct followup of the recent author's paper. In this paper
we continue to analyze approximation and recovery properties with respect to
systems satisfying universal sampling discretization property and a special
unconditionality property. In addition we assume that the subspace spanned by
our system satisfies some Nikol'skii-type inequalities. We concentrate on
recovery with the error measured in the $L_p$ norm for $2\le p<\infty$. We
apply a powerful nonlinear approximation method -- the Weak Orthogonal Matching
Pursuit (WOMP) also known under the name Weak Orthogonal Greedy Algorithm
(WOGA). We establish that the WOMP based on good points for the $L_2$-universal
discretization provides good recovery in the $L_p$ norm for $2\le p<\infty$.
For our recovery algorithms we obtain both the Lebesgue-type inequalities for
individual functions and the error bounds for special classes of multivariate
functions.
  We combine here two deep and powerful techniques -- Lebesgue-type
inequalities for the WOMP and theory of the universal sampling dicretization --
in order to obtain new results in sampling recovery.
