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We show that any complete $k$-partite graph $G$ on $n$ vertices, with $k \ge3$, whose edges are two-coloured, can be covered with two vertex-disjointmonochromatic paths of distinct colours. We prove this under the necessaryassumption that the largest partition class of $G$ contains at most $n/2$vertices. This extends known results for complete and complete bipartitegraphs.  Secondly, we show that in the same situation, all but $o(n)$ vertices of thegraph can be covered with two vertex-disjoint monochromatic cycles of distinctcolours, if colourings close to a split colouring are excluded. From this wederive that the whole graph, if large enough, may be covered with 14vertex-disjoint monochromatic cycles.

Partitioning two-coloured complete multipartite graphs into monochromatic paths and cycles