I am afraid I regard this as an old set of myths - though myths which still keep raising their head. The evidence has been well-researched.
It is rather unlikely that the prim and proper George III had a clandestine marriage, or even an affair, with Hannah Lightfoot
Hannah married Isaac Axford in 1753. He died in 1816, and remarried at the end of 1759, describing himself as a widower. It is not, I think, known for certain when Hannah died, but even if you believe there was a clandestine marriage to George III in 1759, it would be likely to be bigamous on Hannah’s part and hence invalid.
The documents to support the allegation of a clandestine marriage were produced by Olivia (or Olive) Serres, an artist and writer who had fallen on hard times, in the 1810s and 1820s and was seeking to get money from the Crown. She seems to have had an obsession with royalty. Olivia kept changing her story. She started by claiming to be natural daughter of the Duke of Cumberland, George III’s uncle.This story evolved and grew more fantastical over time, with allegations of two other clandestine marriages, one of the Duke of Cumberland to Olivia’s mother, and the other of her mother being the product of a secret marriage between her uncle (a clergyman) and a Polish princess who never visited England. Her claims grew even wilder, with a claim that George III had made her Duchess of Lancaster. Olivia’s husband did not believe any of this. And she also produced documents purporting to demonstrate a secret marriage between George III and Hannah Lightfoot. It seems pretty clear to most experts that these were forged, and ineptly so. They include, for instance, a so-called signature of William Pitt the Elder in which he signs himself as Earl of Chatham some years before he became Earl.
Lavinia seems to have followed in her mother’s fantasising footsteps. In 1866, in an attempt to secure a bequest from the estate of George III, Lavinia and her barrister produced the so-called evidence cobbled together by Olivia, including of the purported 1759 marriage. It is scarcely surprising that the courts dismissed her case.
As for the claim that one George Rex was a son of George III and Hannah Lightfoot, that too experts dismiss. Rex was a family surname, not an indication of close connection with royalty. According to Wikipedia, genetic testing has shown descent from George III improbable. On the DNA testing, see also https://www.genealogy.com/forum/surnames/topics/rex/674/. George was born in 1765, which is almost certainly some years after Hannah’s death.
None of this will of course put off those who want to believe stories like these. They make good fireside tales.