My power went out a few minutes ago (and came back on) in the middle of me using the computer. When this happens, can it have any negative effect on my computer?
Unless it surged enough to fry something inside the computer, then I don't see anything bad happening.
Chances are if something negative happened, you would've noticed by now. Operating systems can become corrupt during immediate power outage but more recently the chances of that happening are slim to none.
the answer to your question: yes it can, but it's probably fine