MacBook Pro shutting down on bootup

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MacBook Pro shutting down on bootup
My MacBook Pro, as of today, shuts down automatically during startup. I believe this to be an issue caused by the limited amount of storage available on the internal harddrive. I've managed to boot from an older backup on my external harddrive, and while I can copy files from the internal to the external, I can't seem to delete any files from it in order to free up some space and bootup properly again. I receive the error "The item can't be moved to the trash because it can't be deleted." Is there any way I can delete files that are on my internal harddrive, while I'm booted from my external harddrive?
Repaly
I do not currently have the means to back up the data on my internal harddrive as my only available external harddrive is the one I'm using to run the older backup from. My original question was is there any way I can delete files that are on my internal harddrive, while I'm booted from my external harddrive?
Do you know if this is possible or not?
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