The context of the Tennyson quote is a eulogy to Prince Albert, in the Dedication of Idylls of the King (and therefore the implied context of a connection to King Arthur?)
Not swaying to this faction or to that;
Not making his high place the lawless perch
Of winged ambitions, nor a vantage-ground
For pleasure; but through all this tract of years
Wearing the white flower of a blameless life,
Before a thousand peering littlenesses,
In that fierce light which beats upon a throne,
And blackens every blot: for where is he,
Who dares foreshadow for an only son
A lovelier life, a more unstained, than his?
Or how should England dreaming of his sons
Hope more for these than some inheritance
Of such a life, a heart, a mind as thine
The first and last parts of this excerpt are clearer than the middle part, at least to me. Tennyson is talking about an exemplary royal who never acted poorly or misused his position, and saying that a life such as his is the best anyone could ever hope for their sons. That leaves the few lines that Gowers quoted and that you're asking about:
Before a thousand peering littlenesses,
In that fierce light which beats upon a throne,
And blackens every blot
I believe this refers to those who might criticise Albert: the "peering littlenesses" being something akin to busybodies, "little" people who "peer" at his life in an attempt to find fault.
With power comes responsibility, and with fame comes visibility: a royal figure will naturally face more scrutiny than someone in a lesser position. Tennyson refers to this fact by a colourful metaphor: the scrutiny to which royals are subjected corresponds to "fierce light" shining upon the "throne" which metonymises the monarchy, and any misdemeanour of theirs which comes to light under such scrutiny corresponds to a "blot" revealed by that fierce light.
For a more modern example, consider Donald Trump. His utterances on Twitter nowadays make international headline news, in a way which they never did before he became president of the USA. His position of power exposes him to more scrutiny - and more criticism - than he would have had otherwise. The "fierce light" of the media shines upon everything he says, and each time he misspeaks it's like a "blot" on his presidential record, made blacker and more visible by the attention accorded to his position.
For the sun to "beat upon" a surface is a common phrase: see, for example, the derived adjective sunbeat. It is perhaps most often heard in the phrase "beat down":
If the sun beats down, it shines very strongly and makes the air very hot
No: a bright light makes dark blots seem darker, not brighter. In the shade, darker and less dark patches are more similar, and the blackest parts don't stand out so much. In bright sunshine, the blackest blots remain black while the rest is lit up, and stand out more by their contrast.
Going back to the Gowers text which refers to the Tennyson quote:
The theory that every act of every official is the act of his Minister is wearing thin. [...] So many people have to read so many official instructions. These offer a bigger target for possible criticism than any other class of writing except journalism, and they are more likely to get it than any other class
What the writer is saying here is that government officials, not just the ministers who lead the government, are liable to receive fierce criticism from the public. To illustrate her point, she contrasts it with the Tennyson quote and says that his "fierce light" (public scrutiny and criticism) no longer shines only upon the apex (the highest echelons of society) but upon the whole pyramid (underlings as well as their superiors).