To begin to read (often implying not finishing); to leaf through or to have something half-read. Frequently written in kana as "読みかける."
ベッドに読みかけの本を置いて眠った。
I left the book I was halfway through on the bed and fell asleep.
仕事の合間に新聞を読み掛けたが、電話がかかってきて中断した。
I started reading the newspaper between tasks, but a phone call interrupted me.
図書館で面白そうな本を読み掛けるだけのつもりだった。
At the library, I only intended to skim a book that looked interesting.