A formal or solemn wish, often directed to a deity or made as a personal pledge. Common in religious contexts, formal requests, and set phrases like 願をかける (make a vow/prayer).
I folded the paper crane, putting my wish into it.
He made a prayer at the shrine, wishing to pass the exam.
She folded a thousand paper cranes with a wish for world peace.
My long-held wish finally came true.
After finishing the pilgrimage, he fulfilled his vow.
祈り refers to the act of praying itself, while 願 is the content of the wish or vow being prayed for.
眼 (がん) is a homophone meaning 'eye' in compounds like 眼科, completely unrelated to 願 (がん), meaning 'wish/vow'.
The kanji 願 combines 原 (source, origin) and 頁 (head), originally meaning 'to turn one's head toward the source' and later extended to 'prayer' or 'wish'. The reading がん is the on'yomi.
written application; written request; petition · written prayer (shrine/temple)
making a prayer to a god or Buddha; votive prayer
at one's request; by request; upon application
supplication; appeal; entreaty; petition
religious vow · prayer (by making a vow)
applying to multiple schools; simultaneous application
fulfillment of a vow
Buddhist vow; arousing the aspiration for enlightenment · prayer; making a vow; praying
long-cherished desire; longstanding wish
great vow; great ambition
praying by proxy; applying by proxy
entreaty; supplication; earnest request
imperial prayer; prayer by imperial command
single application; applying to only one school
single application; applying to only one school
prayer; heartfelt wish
wish; desire
ardent desire; fervent wish
patent application
prayer to a god; making a vow to a deity