peruse
繙 centers on the act of carefully reading through something, like unrolling a scroll to examine its contents. The single meaning 'peruse' captures this idea of thorough, attentive reading.
繙 combines 糸 (thread) with 番, which likely contributes the sound. The thread component suggests the idea of unrolling or unwinding something, which fits the meaning of perusing a scroll or text.
The left side 糸 is thread, and the right side 番 can suggest a sequence or number. Imagine unrolling a threaded scroll page by page to peruse its contents carefully.
For ハン, picture a hand unrolling the scroll: hand -> ハン, and your hand carefully peruses the threaded pages.