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Lumen,
25d. Word for half + abbreviation + word for inlet...all reversed, minus a letter. Perth here isn't in Scotland.
5d. Child is one who stands to inherit.
38d. Def. is 'petty person'. The record is the longer one.

.... I just checked, and the record is actually the shorter one.

Drxx, now I’ve fully understood your post at #30 (apologies for being slow), I agree that it can’t be done as 40 cells without two groups touching at a corner

I don't think this is a very satisfactory puzzle at all. The route to the theme is tortuous. I don't see how the foreign synonyms can be discovered without the help of the internet, which breaches Listener guidelines. More imprortantly, there is no missing name. They are all in the grid in jumbled contiguous arrangements. The only thing that determines the so-called "missing name" is the mathematical consideration. As has been pointed out, the relevant name could be written below the grid without any grid searching since it must be a certain length.

Ever since I've been doing the Listener 'contiguous' has meant touching at edges or corners.

I just double checked mine and I had misspelt the longer name as I had 2 "R''s in the cluster - I now agree with you (Quisling and Loge).

I'm glad you agree on this Quisling - but surely it's the same for the 39 letter version? The optional letter, as I have it, doesn't touch any of the other groups.

I did exactly the same as you, Gitto, before re-checking. Easy mistake to make (well that's my excuse).

I'm struggling to parse 21d and 24d. I'm fully confident of my answer for the latter, I suspect the former is a trap. Doesn't help me that I'm uncertain of the extra letters (I can see two possibilities working, but can get neither to work). Any nudges gratefully received!

Loge, I think that’s right, and certainly the extra letter is unconnected to any other group. I originally thought otherwise, but fresh eyes persuaded me a touching corner for two groups is inevitable. Sorry for the lack of clarity

No need to apologise Quisling. Jumbled items entered in non-conventional shapes lend themselves to potential ambiguities and this the one aspect of the puzzle I didn't like - unlike some others, I thought the thematic connections and gradual revelations made for an enjoyable puzzle.