Hello, Scamper1624.
I understand you are encountering a hiccup when attempting to use a cabled projector to display your iPhone content. I want the chance to help.
The first thing to check would be the compatibility of your adapter. Is it an official Apple adapter, as found in the article I’ve linked below?
About Apple Digital AV Adapters for iPhone, iPad, and iPod touch
The same article also has some great troubleshooting steps as well.
You mentioned that you're using a screen mirroring app on your iPhone. From the description you provided it sounds like it may be unresponsive. Check out the article below for troubleshooting steps to help with the app.
If an app unexpectedly quits, stops responding, or won’t open
If this is not an app that you're trying to use to mirror the screen of your iPhone, are you trying to use AirPlay? Have you gotten to verify this is an AirPlay-compatible projector? You may need to check with the manufacturer for to see if there is an update available if it is. In the meanwhile, I would first make sure that your iPhone and the projector are on the same Wi-Fi network. I’m going to include Apple's AirPlay article below for your reference.
How to AirPlay video and mirror your device's screen
I hope you find this helpful. If you are still unable to connect with your projector, please reply with as many non-personal details as possible. The more information you can provide, the more assistance can be given.
Best Regards.