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As you are in the UK then if you bought your phone in the UK then we have some of the strongest consumer laws on our side, If this is a defect which it sounds like it is if what you say is true and the screen is coming away and not broke ie: cracked then you should be able to have this repaired under warranty but your 1st port of call is not Sony but the place you bought it from unless it was direct from Sony - If from a store and they refuse to repair then ask for a Deadlock letter then 99.9% of the time they will back down and repair and I'm speaking from lots of experience and even if they give you one then have it inspected else where and get a report from them stating it's a defect then they will have to fix it and reimburse you any costs you have incurred
Contract or bought EE is still the place to go and EE is probably in my opinion the best at fixing phones quickly and hassle free unlike other shops and networks, As long as the screen isn't cracked then it should be fixed - Good thing about EE they do give you a loan phone but it's a bog standard old thing
I have exactly the same issue, with no damage or drop the top left corner of my screen is lifting off away from the body of the phone. Have sent to Sony for repair. This will be the second repair, first the SIM and SD Card covers had to be replaced as they didn't close properly, and now the screen isn't fixed to the chassis properly. All in all I feel rather let down by the build quality of the phone. First Sony mobile I've had for many years and very disappointed that within the first 6 months there have been 2 manufacturing defects.
I just used the support tab on the Sony Mobile website to create a repair order. Took about 7 days to be fixed, i think the initial production run was faulty.