Translation:This book does not have the second page.
In British English we would say 'hasn't got a second page'. 'doesn't have' has a different meaning (repetitive) - e.g. not very often
Wouldn't "This book lacks the second page" also be a pretty normal translation?
Though my preferred translation is "This book has no second page", I experimentally tried "...does not have its second page" (implied its, cf. implied my in the exercise where "ciocia" = "my aunt"), but Duo didn't like it.
Am I completely wrong in this case?
Well, I'm surprised that you would say "its", even given the fact how often English uses possessives especially in comparison with Polish... but I trust your judgement, if you say it makes sense as an English sentence, then I can add it, it sure means the same.
I think 'its second page' makes sense here because even if you ripped a load of pages, starting with p2, out of a book, it would still have 'a' second page, it's just that that second page might be p22 in the printed numbering and not p2, as it should have been.
Though of course even-numbered pages are always the verso, so it would be tricky not to rip out page 1 too!