  We study the interplay between the notions of $n$-coherent rings and finitely
$n$-presented modules, and also study the relative homological algebra
associated to them. We show that the $n$-coherency of a ring is equivalent to
the thickness of the class of finitely $n$-presented modules. The relative
homological algebra part comes from the study of orthogonal complements to this
class of modules with respect to ${\rm Ext}^1_R(F,-)$ and ${\rm Tor}_1^R(F,-)$.
We also construct cotorsion pairs from these orthogonal complements, allowing
us to provide further characterizations of $n$-coherent rings.
