。
She earnestly hopes for her son's success.
Literal
She [topic-は] son [genitive-の] success [object-を] earnestly is-hoping-for.
切に ('earnestly, ardently') is a literary adverb built from 切 (the noun meaning 'cutting / earnestness') + に; the kanji-and-に combination creates a slightly elevated register. Common collocations: 切に願う / 切に望む — paired with verbs of hope and prayer, often in formal letters or speeches. The present-state 望んでいる marks the hoping as ongoing, sustained over time.