We introduce and study the family of sets in a finite dimensional Euclidean space which can be written as the Minkowski sum of a compact and convex set and a convex cone (not necessarily closed). We establish several properties of the class of such sets, called Motzkin predecomposable, some of which also hold for the class of Motzkin decomposable sets (i.e. those for which the convex cone in the decomposition is requested to be closed), while others are specific of the nex family.