to ask for; to apply for; to make an application for; to file a request for; to submit (e.g. one's resignation)
上司に退職を願い出た。
I went to my boss and asked to resign.
異動を願い出ているところだ。
I'm in the process of requesting a transfer.
正式に休暇を願い出る手続きをした。
I went through the formal process to apply for leave.