Short answer: maybe — but don’t count on it. Most monitoring apps (mSpy, uMobix, etc.) can log messages, notifications, or — on rooted/jailbroken phones — take screen captures or record screens. If a post is deleted seconds after upload, it’s only recoverable if the monitoring tool actually captured it in real time (or the phone had automatic backups). Once the user deletes it from Instagram’s client and no screenshot/video exists, it’s effectively gone to you.
Don’t leave this to luck. Enable strict parental controls, automatic backups, push notifications for accounts you monitor, and run regular device checks. If you’re testing tools, try a controlled trial on the device first to confirm what they actually catch.