This is about checking up on your publisher. Does anyone ever do this? I've no idea. But just in case you want to...
18. Examination of Publishers' [sic] Records
The Author or the Author's authorised representative shall have the right upon written request to examine the records of account of the Publishers insofar as they relate to the sales and receipts in respect of the Work, which examination shall be at the cost of the Author unless errors exceeding £10 (ten pounds) to the Author's disadvantage shall be found, in which case the cost shall be paid by the Publishers.
This means that if you think they are fiddling your royalties, or are incompetent, you can look at their records relating to your book for £10. If you are right, and they owe you more than £10, you don't have to pay the money. This clause doesn't actually say they will pay what they owe you, I but I assume they will. Now, if the publisher *is* fiddling the royalties, they are hardly going to leave a paper trail you can pick up so this won't do you any good. Perhaps I am being naive, but I think worrying that the publisher is withholding royalties ranks along with the 'will they steal my idea?' terror that besets unpublished writers. It must cost more to fiddle the accounts than to pay the royalties due, surely? If they have got them wrong because they are incompetent - well, checking the records might do you some good. It might at least show you it's time to find another publisher.
Incompetence is the more likely reason for the royalty statement being wrong. Usually, if you query a royalty payment and ask them to check, they will do so. It's not really necessary to go this £10 route. I think this is a rather boring clause (unless you are totally paranoid) that gives little potential for a fight.
Notice that it is specifically for 'the Work' - ie the current book. If you publish a lot with the same publisher, I'd argue you want to be able to look at the records for all your books for the same £10.
The Society of Authors does random spot checks on royalties and I suspect that keeps publishers on their toes anyway. Personally, I'd rather write another book than bother with faffing around in someone else's accounts. It's bad enough faffing around in my own accounts. I suppose now I've said that, all my publishers will start fiddling my royalties with impunity, confident that I am never going to check...