For $a>0$, what is $\int_1^\infty dy \, \exp(-ay) (y^2-1)^{-1/2}y^{-2}$?

The proposed integral is a Bickley-Naylor function. It can be defined as \begin{equation} \operatorname {Ki} _{n}(x)=\int _{1}^{\infty }{\frac {e^{-xt}\,dt}{t^{n}{\sqrt {t^2-1}}}} \end{equation} For $n=2$, which is the case of interest, \begin{align} \operatorname {Ki} _{2}(x)=1-{\frac {\pi }{2}}x-&{\frac {x^{2}}{2}}\left(\gamma +\ln \left({\frac {x}{2}}\right)\right)\sum _{k=0}^{\infty }{\frac {(x^{2}/4)^{k}}{k!(k+1)!(2k+1)}}+\\ &+{\frac {x^{2}}{4}}\sum _{k=0}^{\infty }{\frac {(4k+3)(x^{2}/4)^{k}}{k!(k+1)!(2k+1)^{2}}}+{\frac {x^{2}}{2}}\sum _{k=1}^{\infty }{\frac {(x^{2}/4)^{k}\Phi (k+1)}{k!(k+1)!(2k+1)}} \end{align} where $\gamma$ is the Euler constant and $\phi(k+1)=1+1/2+1/3+\cdots+1/k$.

and for $x\to\infty$, $\operatorname {Ki} _{2}(x)\simeq e^{-x}(\pi/\sqrt{2x})$.

The Wikipedia link provides several properties and references.