CSP sparsification, introduced by Kogan and Krauthgamer (ITCS 2015), considers the following question: how much can an instance of a constraint satisfaction problem be sparsified (by retaining a reweighted subset of the constraints) while still roughly capturing the weight of constraints satisfied by {\em every} assignment. CSP sparsification captures as a special case several well-studied problems including graph cut-sparsification, hypergraph cut-sparsification, hypergraph XOR-sparsification, and corresponds to a general class of hypergraph sparsification problems where an arbitrary $0/1$-valued {\em splitting function} is used to define the notion of cutting a hyperedge (see, for instance, Veldt-Benson-Kleinberg SIAM Review 2022). The main question here is to understand, for a given constraint predicate $P:Σ^r \to \{0,1\}$ (where variables are assigned values in $Σ$), the smallest constant $c$ such that $\widetilde{O}(n^c)$ sized sparsifiers exist for every instance of a constraint satisfaction problem over $P$. A recent work of Khanna, Putterman and Sudan (SODA 2024) [KPS24] showed {\em existence} of near-linear size sparsifiers for new classes of CSPs. In this work (1) we significantly extend the class of CSPs for which nearly linear-size sparsifications can be shown to exist while also extending the scope to settings with non-linear-sized sparsifications; (2) we give a polynomial-time algorithm to extract such sparsifications for all the problems we study including the first efficient sparsification algorithms for the problems studied in [KPS24].