The paper under review introduces two classes of modules, called weak $n$-injective and weak $n$-flat, which are the generalizations of $n$-FP-injective and $n$-flat modules, respectively. In order to be more precise, let $R$ be a ring. An $R$ module is called super finitely presented if it has a projective resolution consisting of finitely generated (projective) modules. An $R$ module is called weakly $n$-injective ($n$-flat) provided that it is injective (respectively, flat) with respect to all super finitely presented modules of projective dimension less than or equal to $n$. In the paper, it is shown that the class of weakly $n$-injective ($n$-flat) modules is the right (left) component of a hereditary (perfect hereditary) cotorsion pair.