Is it correct to mix tripod and handheld footage?
You're asking for an opinion, because there's no rule book to consult here. So, IMO...

'Pro' movies are shot with all manner and mixtures of techniques, including dolly, mount, rig, crane, Steadicam, fixed, zoom, drone etc. I think many people would find a short film that mixed tripod with deliberate "shaky cam" to be a bit jarring without a good cinematic justification, but just mixing IS handheld with tripod should be no sin. Story trumps technique anyway -- if your viewers are watching how the camera moves, your story isn't compelling -- or they're film students. (-: