Hello Philip_R,
Thanks for posting in the Apple Support Communities.
We understand the audio is delayed when you're using Duolingo with the headset only, right?
If we have this correct, let's confirm that you have the latest iOS and iPadOS versions installed to see if that may help with clearing up what's occurring. If an update is available, be sure to complete a backup of your iPhone before installing this. Check out the steps below for assistance, and you can find more details here: Update your iPhone or iPad - Apple Support
Update your iPhone or iPad wirelessly
1. Back up your device using iCloud or your computer.
2. Plug your device into power and connect to the internet with Wi-Fi.
3. Go to Settings > General, then tap Software Update.
4. If you see more than one software update option available, choose the one that you want to install.

5. Tap Install Now. If you see Download and Install instead, tap it to download the update, enter your passcode, then tap Install Now. If you don't know your passcode, learn what to do.
If you have the latest updates installed, check for an app update to see if that may improve this. Since it seems to be isolated to Duolingo only, check out the steps of the following resource for an update: How to manually update apps on your Apple device - Apple Support
We hope this is able to help with what you're experiencing. If not, confirm which steps you've taken so far and provide any additional details to help us understand this better. We'll provide you with the best next steps to take.
Have a great day!