Ah my bad, I mistook your problem as that with lenses. No, if the reflecting surface is the surface of the mirror itself and no penetration is possible, its thickness won't matter to the reflected wave. Note: up to now I'm assuming your definition of mirror thickness as the distance between the mirror's vertex and its back.
I think you should recheck again how this source defines the thickness of a mirror.
"We will see that the
radius of curvature, which is a length is exactly
the reciprocal of the curvature", quoted from
http://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter15/section03.html
The curvature of a lens is something you can control during the fabrication, you can have thick lenses with small or big curvature.
May be you meant depth of field instead of sharpness. If you really mean sharpness, it's the other way around, restricting the aperture to small diameter will reduce the image's resolution.