I had not heard of the "40 cycle amplification" thing until I saw your post just now. Therefore, what I'm posting below just comes from a few minutes of Googling. We'll really want actual medical folks like
@growlers,
@Biff84,
@ghostguy123,
@Terminalxylem,
@gianmarco,
@ProstheticRGK, and
@jm192 to weigh in.
...
"40 cycle amplification" is part of the procedure for the polymerase chain reaction test (PCR). PCR tests are used for many kind of viruses, including COVID.
Here, "amplification" just means copying of a small piece of DNA over and over until there is enough to more easily (?) analyze. There appears to also be a way to derive copies of the virus's RNA from this same procedure, but I don't know the science behind that.
Anyway,
this article suggests that a range of amplification cycles between 20 and 40 is common. Under 20 doesn't seem to yield enough material to be useful, and over 40 can sometimes be too tenuous (e.g. uninfectious viral particles get overrepresented in a sample, leading to false positives).
This 2012 article is above my head, but if you CTRL+F for the word "cycle", you'll find lots of side discussion about what's the "sweet spot" number of cycles for this kind of test or that kind of test. Some commentators complain how a 30-cycle test and a 35-cycle test of the same sample yields different results and are asking for help from other researchers to interpret the results.
Additionally, number of cycles has been an issue for the development of rapid-results PCR tests --
some reading here in this 2011 article.
I haven't seen anything about WHO's take on COVID testing amplification cycles. But on the surface, it seems to this layman that WHO recommending a lower number of cycles from a given standard can make sense. It could be true that 40 cycles is too much and the resulting samples are sullied by the presence of non-infection viral particles. But it's all something on which we'll want expert consensus.