“Memorabilia,” the person on the door says, passing out a commemorative ticket welcoming fans to Cruz Beckham’s show in “Cardiff, England”. It’s an inauspicious start, particularly in a building that’s a decades-old monument to Welsh-language culture, but once the house lights go down, there’s enough to suggest that, even if he’ll never make it as a geography teacher, he could give this music thing a proper lash.

It would be easy to go in studs-up on Beckham, the youngest son of David and Victoria. He’s already a tabloid fixture, his profile supercharged by the recent intra-family beef that has opened up new seams in the content mines, and his every move might duly be viewed with cynicism. His desire to play clubs, to “do it the right way”? The machinations of a nepo-prince currying favour with the masses.

And yet, there is an earnestness about him on stage, an unstudied enthusiasm, that undercuts this narrative. “Bloody hell, wow,” he says after one song, mussing his hair as the half-full room claps. “I promise I’m not an arsehole,” he sings during Tie Don’t Fit, his voice like an off-the-shelf Lee Mavers. A little on the nose, perhaps, but Beckham’s music is also all over the place in a manner that reflects the difficulty in figuring things out next to the pop of flashbulbs.

His first shows with his backing band, a three-piece called the Breakers, were only a few months ago, and this is his second headline gig. He’s woodshedding ideas in public, bouncing from Lick the Toad’s landfill indie to solipsistic ballads, but in the Beatles-esque pivot at the heart of For Your Love – written alongside Justin Raisen and Lewis Pesacov, cool-guy producers who’ve worked with Kim Gordon, Sky Ferreira and Best Coast between them – he uncovers psych-pop worthy of further investigation.

It’s there again in Wear and Tear, which throws dubby bass into a baggy mix, and Better Times, a sunny could-be-hit where Beckham proves himself to be a characterful guitarist, peeling off a fuzz-bomb solo that’s nimble and melodically savvy. A good-looking kid who can play? It’s worked before.

• Cruz Beckham plays Exchange, Bristol, 28 February; then touring UK and Europe until 27 March