$$\int_{-\infty}^{-(-a+\frac{b}{y})^{1/2}}\frac{e^{-x^2}}{x}dx=\tfrac{1}{2}{\rm Ei}\,(a-b/y)=$$
$$\qquad\qquad=\tfrac{1}{2}(a-b/y)^{-1}\exp(a-b/y)\sum_{n=0}^\infty\frac{n!}{(a-b/y)^n}$$
$$\qquad\qquad=-\frac{y }{2 b}e^{a-\frac{b}{y}}\sum_{n=0}^\infty (-1)^{n}n!(1-a)^n(y/b)^n$$