Which sentence is better? If none, please make one.

1) She held a novel in her hand, reading it absent-mindedly.

2) A book in hand, she cannot concentrate on what she's reading.

3) With a book in hand, she read it absent-mindedly.

4) She's holding a book reading, unseeingly.
Sentence 1 is fine.

Sentence 2: you would normally omit the article, i.e.: 'Book in hand, she...'

Even then the sentence sounds awkward, because the clause after 'book in hand' would normally deal with a different activity. 'Book in hand, she chased him round the garden.'

Sentence 3 isn't quite right: again, the main clause will not normally refer directly to the 'with a...' clause, e.g.

'With a book in her hand, she ran down the stairs.'
'With a song on her lips, she dug the garden.'
'With hatred in her eyes, she stuck the knife in his belly.'
The explanation was cool.
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Indeed - subspecies UK.