  A Weierstrass fibration is an elliptic fibration $Y\to B$ whose total space
$Y$ may be given by a global Weierstrass equation in a $\mathbb{P}^2$-bundle
over $B$. In this note, we compute stringy Hirzebruch classes of singular
Weierstrass fibrations associated with constructing non-Abelian gauge theories
in $F$-theory. For each Weierstrass fibration $Y\to B$ we then derive a
generating function $\chi^{\text{str}}_y(Y;t)$, whose degree-$d$ coefficient
encodes the stringy $\chi_y$-genus of $Y\to B$ over an unspecified base of
dimension $d$, solely in terms of invariants of the base. To facilitate our
computations, we prove a formula for general characteristic classes of blowups
along (possibly singular) complete intersections.
