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.

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