I am unable to back up phone because I cannot press the trust this device button as the screen is broken

You should be prepared for the phone to be completely erased if it is fixed. You'll have to use an older backup. If you've been backing up to iCloud, plug it in and let it backup overnight. If you've only been backing up to a computer, there is likely little that you can do.

Regardless of the repair being done, it's always best to prepare for complete data loss. That's all I'm saying. You should be prepared to accept that the data may be lost. Technically, there should not be any issue, but anytime you open a device for repair there is a possibility.

It is also important to realize that Apple’s terms of service do not guarantee that any service will not result in data loss. Apple’s accepts no responsibility for anyone’s data when they service a device.

It is always up to device owners to safe guard their data. So even for repairs where data loss is unlikely given the nature of the repair, if it should occur, Apple is not responsible for safe guarding anyone’s data.