Finally into double digits on the clauses - but we're still not half way through the contract. This is a pleasant clause as it's about what you get rather than what you have to do:
10. Author's Free Copies
On publication, the Author will immediately be sent x (x) presentation copies of the first edition, direct to her at Stroppy Author's address and 1 (one) copy of any subsequent edition of the Work and shall have the right to purchase further copies at trade terms for personal use but not for resale. On publication of the first edition, the Agent will also immediately be sent one complimentary copy, to Stroppy Author's Agent.
The copies of your books they give you on publication are called contractual copies. The main thing to argue here is about how many you get. It can range from 1 to 24. The most I have ever had is 18; the least is 0. (Yes, I have one book out there that I have never even seen.) However many they offer you, ask for more - unless you really can't see yourself giving away copies the book. There's no point having them just for the sake of it, they clog up the bookshelf space that could be occupied by other books that you actually want to read. Eight or ten is a good number. But see the bit below about school visits before deciding.
It has to be said that not all publishers are very diligent about sending your contractual copies. You might have to nag for them. The publisher might not even tell you the book has been published. I have occasionally bought my own book on Amazon because I've needed a copy to take somewhere/give someone and the contractual copies haven't arrived.
If there are people you have to give copies to because they helped you tell the publishers they will have to send copies to these people. (This is people who have helped you with information - not people who made your coffee and wiped your fevered brow. And not members of your family who read your early drafts [shudder - never show early drafts]). This is not really a contractual issue, but it's worth telling them at this stage so that if they say 'you will have x copies, send them one of those', you can argue here for the number to be increased. Publishers will usually give you extra free copies after publication if you are giving them to people they consider sufficiently interesting or influential, too.
As for copies of subsequent editions, in my experience they hardly ever turn up. It is worth asking what this means - does it mean you will get the paperback? Does it mean you will get a copy of every foreign edition? Does it mean that if they re-use your text in another book (which they may do if it is non-fiction and you don't retain copyright) that you will get a copy of that other book? Insist on all these. It may seem rather pointless, but if you have a copy of every edition, you can register them all for PLR. Otherwise, extra versions tend to come out and you don't know about them so you lose out on the PLR for those editions. It's OK to explain this reasoning to the publisher - if they can see you have a good reason for wanting the copies, they may be more likely to send them. They may. But don't hold your breath. It's often worth checking on Amazon every now and then to see which new editions have come out. You can then chase them up if you wish.
Make sure there is a copy for your agent. Or don't. If your agent reads the contract and doesn't notice if they are not down to get a copy, they don't deserve one. Nor do they deserve to stay as your agent.
The bit about not selling your copies may be of no consequence to you, but many children's authors make visits to schools and like to sell copies of their books at these visits. Personally, I do neither school visits nor selling, so I would be grateful if anyone who does could add in the comments any further advice on how this relates to the contract. As I understand it, most authors buy books at discount from their publisher and sell them on school visits. Maybe this is separately negotiated, or maybe the publishers just turn a blind eye. To do so would breach this contractual term, so if you plan to visit schools and set up your stall, you might want to challenge this clause, or at least clarify how you get discounted books to sell. (I often find it's cheaper to get my books on Amazon, often from re-sellers, when I need extra copies to give away.)