How do I find out if I can fix my display

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If your iPhone has a history of Screen replacement / Repair / Service from an un-authorized service provider then the only option that you have is to take the device to...

Take your device to an Apple Retail Store or Apple Authorized Service Provider, or contact Apple Support:

Find Locations - Location Selector - Apple

Contact Apple for support and service - Apple Support 

Get a Genius Bar Reservation and Apple Support Options - Apple

Contact Genius Bar.

Manage Reservations - My Reservations - Apple

Parts and service history

With iOS 15.2 and later and iPhone XR, iPhone XS models, and later—including iPhone SE (2nd and 3rd generation)—you can go to Settings > General > About for the parts and service history of your iPhone. You can also find out if the battery has been replaced. If the replacement was done using genuine Apple parts and processes, "Genuine Apple Part" will appear next to Battery.