Application, Applying
noun, verbal noun
When you say humbly that you're requesting something, you're putting forward an application. 申請 refers to the process of officially applying for approval or permission, like ビザの申請 (application for a visa), ライセンスの申請 (application for a license), or 休暇の申請 (application for leave/vacation).
しんせい
This is a jukugo word, which means on'yomi readings from the kanji. You already know this reading for 請, but not for 申, so here's a mnemonic to help you: Today you finally submitted your application for a new shin (しん). Your old one had to be painstakingly removed after you banged it against a table. Anyway, congrats on the application for your new shin!
しまった！ビザの申請を忘れてた。
Oh no! I'd forgotten to apply for a visa.
申請手続きに時間がかかってるんです。
The application process is taking quite a while to complete.
ようやく許可申請が受理されました！
At last, the application for permission has been accepted!