I remember that in original Basic Roleplay you can increase your power stat through use but how does it work in Call of Cthulhu 5e? My friend mentioned having to best someone in mental combat but I can't seem to find a reference to it.
What you're talking about is winning a contest of POW vs POW (see The Resistance Table, pp. 49-50). In RuneQuest this is relatively easy to arrange; in Call of Cthulhu it's rather less so, though casting the Shrivelling spell (p. 214) is the traditional approach.
See the box on p. 93 ("How Sorcerers Get That Way") for details on this and other means of improving POW.