Moravec, Harris noisy window
I've added a link to the paper you're quoting in the question.

The issue is that both the Moravec corner detector, and the change to it suggested first in that paper:



use the derivative of the image. ANY difference operation will tend to enhance higher frequency noise in the image.  The idea with the Gaussian smoother is to reduce this effect.  The two operations (Gaussian smoothing and difference operation) are often combined into a single operation: the difference of Gaussians.

As the paper suggests, the anisotropy of the Moravec detector is why the differential is used, rather than the up, down, and diagonal differences in the Moravec algorithm. The smoothing is introduced to counter the noise issue.