Can someone explain what a boxout is? Or show an example on an A4 page
I've never heard of that expression. I suggest you contact the publisher for clarification. Also confirm whether the half page is upright or horizontal.
Yeah, I understand that – such as a pull quote – but I don't see a publisher specifying that for you to produce artwork for a halfpage press advertisment to be included on an A4 page. Obviously you can include a pull quote in your design, but that's a matter for you and your customer, not the media publisher.
I'm wondering if perhaps this is a black and white publication and the boxout might be referring to a section of the ad that is reversed out of a black box for emphasis.
We could use some context as to how the publication is using the term in what they are telling you (the specs).