In many graph problems, like Longest Induced Path, Maximum Induced Forest, etc., we are given as input a graph G and the goal is to compute a largest induced subgraph G[F], of treewidth at most a constant t, and satisfying some property P. Fomin et al.  proved that this generic problem is polynomial on the class of graphs Gpoly, i.e., the graphs having at most poly(n) minimal separators for some polynomial poly, when property P is expressible in counting monadic second order logic (CMSO). Here we consider the class Gpoly + kv, formed by graphs of Gpoly to which we may add a set of at most k vertices with arbitrary adjacencies, called modulator. We prove that the generic optimization problem is fixed parameter tractable on Gpoly + kv, with parameter k, if the modulator is also part of the input. The running time is of type O(f(k + t, P) · nt+5 · (poly(n)2)), for some function f.