  We compute the first-order $\alpha'$-corrections to a general family of
supersymmetric heterotic backgrounds which describes a rotating superposition
of a fundamental string and a momentum wave. We focus on a particular solution
within this family that gives raise, upon dimensional reduction to five
dimensions, to a black ring with two charges and one dipole that has a singular
horizon with vanishing area in the supergravity approximation, better known as
small black ring. We show that the singularity at the horizon persists even
after the first-order $\alpha'$-corrections have been taken into account. This
implies, contrary to what has been argued in the literature, that
higher-derivative corrections do not resolve the singular horizon of the
5-dimensional small black ring, at least at first-order in $\alpha'$.
