Hey there PeteBr,
Thanks for reaching out to the Apple Support Communities. To get started, are you only experiencing issues with using the search function in the Notes app? For example, if you make a search after swiping down from the middle of the Home screen, does the same issue persist?
If not, try restarting your iPhone with the steps shown here: Restart your iPhone - Apple Support
How to restart your iPhone 6, 7, 8, or SE (2nd or 3rd generation)
1. Press and hold the side button until the power-off slider appears.
2. Drag the slider, then wait 30 seconds for your device to turn off. If your device is frozen or unresponsive, force restart your device.
3. To turn your device back on, press and hold the side button until you see the Apple logo.
While restarting may seem like a simple step, it can often help with resolving some unexpected issues like this.
Let us know if that's able to help.
Have a great day!