This is an integral in Oberhettinger's Table of Fourier Transforms (page 27, with $r=\sqrt{a^2+x^2}$). It is not a very helpful result, but it does suggest that a fully closed-form expression will not be forthcoming.
There also seems to be some typo in the formula, a numerical check does not match. There is an error in both branches, for $y<b$ and for $y>b$. For $y<b$ a square root is missing in the argument of the Bessel $Y_0$ function,
$$\int_{0}^\infty \frac{\sin\left(b\sqrt{a^2+x^2}\right)}{a^2+x^2}\cos yx\,dx =\frac{e^{-a y} \,\text{Ei}(a y)-e^ {ay} \,\text{Ei}(-a y)}{2 a}$$ $$\qquad\qquad-\frac{\pi}{2} \int_y^b Y_0\left(a \sqrt{t^2-y^2}\right) \, dt,\;\;0\leq y<b,$$