From the introduction: Throughout this paper \(R\) will denote a commutative noetherian ring (with a non-zero identity). We shall follow \textit{I. G. Macdonald's} terminology [see Symp. Math. 11, Algebra Commut., Geometria, Conf. 1971/72, 23--43 (1973; Zbl 0271.13001)] concerning secondary representation. So whenever an \(R\)-module \(L\) has a secondary representation, then the set of attached primes of \(L\), which is uniquely determined, is denoted by \(\text{Att}_R(L)\). \textit{H. Ansari-Toroghy} and \textit{R. Y. Sharp} [Proc. Edinb. Math. Soc., II. Ser. 34, No. 1, 155--160 (1991; Zbl 0731.13005)] showed that if \(M\) and \(E\) are respectively a finitely generated and an injective \(R\) modules, then \(\Hom_R(M, E)\) has a secondary representation. Also they described \(\text{Att}_R(\Hom_R(M,E))\) in terms of \(\text{Ass}_R(M)\) and a certain set which is uniquely determined by \(E\). In this paper we will show that the above arguments are still true under a weaker condition when \(M\) is an \(R\)-module with the property that its zero submodule has a primary decomposition and \(E\) an \(R\)-module which is injective relative to \(M\). In this case, the functor \(\Hom_R(-, E)\) is not exact in general. We recall that \(E\) is injective relative to \(M\) (or \(E\) is \(M\)-injective) if and only if for any submodule \(N\) of \(M\) (up to embedding), the homomorphism \(\Hom_R(M, E)\to\Hom_R(N, E)\) is epic.