The input to a PPNC instance is integers n and p, and a non-negative real weighting of the edges of the clique Kn on the vertex set {1,..., n}. We are asked to find a set of p disjoint cycles spanning {1,..., n} and subject to this such that the sum of the weights of the edges is minimized. We provide an efficient approximation algorithm for the metric version of this problem which has an approximation ratio of 4 if p ≤ n/5 and an approximation ratio of 51 for larger p. For p > n/5, our algorithm uses a subroutine which approximately solves the Mod 3 T-join With Slack problem. The input to an instance of Mod k T-join with Slack consists of integers n and B, a non-negative weighting of the edges of the clique Kn, and a label l(v) from {0,1,..., k - 1} on each vertex of Kn. We are asked to find the minimum weight spanning forest F from amongst those satisfying ∑T∈F((∑v∈V(T)l(v)) mod k) ≤ B. If k = 2 and B = 0 this is the well-studied T-join problem which can be solved exactly in polynomial time.