What state is it in when you connect to the PC? File transfer via MTP requires that the Android operating system is running, so if it won't boot I'd not expect you to be able to see the filesystem.
I don't know what the procedure and tools for flashing a Sony are (each manufacturer has its own way of doing this), but I'd guess you would have to boot it into fastboot or download mode (some tools would connect to the phone when it is running Android and tell it to boot into the correct mode, but that sounds unlikely to work in this state). Which raises the question of whether the phone will boot into those modes, or recovery mode? If it is stable in one of those modes there's a chance something can be done.
If it was bootlooping (constantly restarts part way through the boot up) I'd suggest going into recovery mode and trying a factory reset, but this sounds a bit worse than that. You could still try, but I'm not optimistic (of course you'd lose any data that aren't backed up, so if there is anything important there I'd keep this as a last resort).