You should be able to power a 2.5″ drive using the Pi’s USB port. Depends one the drive’s power consumption though I guess.
Does your config.txt contain max_usb_current=1 ?
Could maybe be the PSU also. Any way of testing the output? It could be a cheap Chinese thing that doesn’t actually provide a good clean signal and/or the rated amperage. Have you got another PSU you could try?