The sight proper looks a lot like a "Central". My WAG is that the extra bit of glass is probably a "corrective" lense placed to that the shooter can achieve correct sight focus without having their normal spectacles sliding down their sweating nose.
As the human eye ages, it gets bigger by adding extra layers like an onion. This especially happens to the lense. This is why, as you age, it is harder to rapidly "pull focus" from near to far and vice-versa. The catch is that the "larger", more rigid eye-ball and lense assembly tends to favour "distance" vision and the little annular muscles that contract and "bulge" the lense for "close" vision have to struggle harder with each passing year. Having to hold the newspaper at arms length is a bit of a giveaway.
One of the basic rules of (iron-sighted) marksmanship is to focus on the front-sight, thus, older shooters may require some assistance. Looking through a small aperture certainly helps to some extent, but a corrective lense will tidy things up a bit more.